Different people have a different understanding of the concept of the Holy Trinity. The idea also has developed over time. It first became church doctrine in the fourth to fifth century after Christ's birth. It became more radical with time, the further we are removed from the time of Christ. Today many will say that Jesus and God are one, or at least that Jesus and God are of equal power.  It has also become more commonly accepted, no major church  is in disagreement anymore, since the Adventists have accepted it in the 20th century.
The followers of this doctrine have used many symbols to reflect the difficult matter of one substance but three views (persons), examples are the equal sided triangle, a leave like that of clover, three smaller overlapping circles in one large circle, etc.
The teaching of Trinity is part of many songs and thrown in at will (without logically connection to the previous or following thoughts) in services, at times it seems just because of doctrine, just like in politically correct speeches and essays certain statements are made whether called for by context or not.
As I said, a long time ago the church was for centuries quite divided about whether God, Jesus and the Holy Ghost were all equally god. For some reason in the end most of the church leadership opted for the Holy Trinity version. As you know at that time, like for the next 1000 to 1200 years the rulers determined the faith of the population (“cuius regio, eius religio“), maybe or likely it was as simple as that for the trinity doctrine as well.
We all know that the church has erred in many ways, for example: 
Therefore the question of “Yes” or “No” to the concept of trinity can not be considered to be just one of church doctrine, but it needs to be a pursuit of the truth, driven by the desire that fellow men are not lead astray, becauseThe first commandment reads:
And God spake all these words, saying, I am the LORD thy God, which have brought thee out of the land of Egypt, out of the house of bondage. Thou shalt have no other gods before me.
What if we are in conflict with the first commandment by accepting a doctrine not supported by the Bible. 
You heard me say, ‘I am going away. And I am coming back to you.’ If you loved me, you would be glad I am going to the Father. The Father is greater than I am.
Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I.
This quote reminds me of:
I baptise you with water for repentance. But after me will come one who is more powerful than I, whose sandals I am not fit to carry. He will baptise you with the Holy Spirit and with fire.
Jesus like John, the baptist, said he is serving somebody higher, again distinguishing clearly.
Jesus prayed that not “not my will, but thine, be done“, clearly distinguishing between his own will and that of his father. (One substance of equal knowledge and perfect goodness would lead to one will, would therefore make Jesus and God have one will even if they appear to be two different persons). The fact that there is only one incidence reported with a difference of will between father and son, does not change anything about the implication of Jesus' statement.
Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.
He said, “Father, if you are willing, take this cup of suffering away from me. But do what you want, not what I want.”
Jesus said to Mary Magdalene shortly after his ressurection:
Do not hold on to me, for I have not yet returned to the Father. Go instead to my brothers and tell them, 'I am returning to my Father and your Father, to my God and your God.'
Most of the time Jesus speaks of God as his father as of his God but more than once also of his God. If Jesus was God and had a God, then there would be at least two Gods. But that would clearly contradict Jesus - as well as all of the Bible:
Now this is eternal life: that they may know you, the only true God, and Jesus Christ, whom you have sent
If the Father, the person Jesus is addressing in his prayer, should be the only true God – as Jesus says – then Jesus who according to the concept of the holy trinity is a person other than the Father, can not also be God!
Jesus says in his epistle to the church in Philadelphia:
I am coming soon. Hold on to what you have, so that no-one will take your crown. Him who overcomes I will make a pillar in the temple of my God. Never again will he leave it. I will write on him the name of my God and the name of the city of my God, the new Jerusalem, which is coming down out of heaven from my God; and I will also write on him my new name.
Even after his resurrection, glorification and being seated at the right hand of God – Jesus speaks of his father as of his God!
It is clear from this statement made by Jesus that he in contrast to the almighty God he does not have this power on his own.
Jesus said after his resurrection:
All authority in heaven and on earth has been given to me.
This is similar to all power over Egypt having been given by the pharaoh tp Joseph. Yet the power was still the Pharaoh's (Joseph was also a subordinate).
This shows how much power and honour was bestowed on Joseph.
And Pharaoh said unto Joseph, See, I have set thee over all the land of Egypt. And Pharaoh took off his ring from his hand, and put it upon Joseph' hand, and arrayed him in vestures of fine linen, and put a gold chain about his neck; And he made him to ride in the second chariot which he had; and they cried before him, Bow the knee: and he made him ruler over all the land of Egypt. And Pharaoh said unto Joseph, I am Pharaoh, and without thee shall no man lift up his hand or foot in all the land of Egypt.
Jesus' statement is also repeated by Paul / the writer of the letter to the Ephesians:
I keep asking that the God of our Lord Jesus Christ, the glorious Father, may give you the Spirit of wisdom and revelation, so that you may know him better.
Similarly John speaks of the God and father of Jesus Christ:
John, To the seven churches in the province of Asia: Grace and peace to you from him who is, and who was, and who is to come, and from the seven spirits before his throne, and from Jesus Christ, who is the faithful witness, the firstborn from the dead, and the ruler of the kings of the earth. To him who loves us and has freed us from our sins by his blood, and has made us to be a kingdom and priests to serve his God and Father - to him be glory and power for ever and ever!
For he hath put all things under his feet. But when he saith, all things are put under him, it is manifest that he is excepted, which did put all things under him. And when all things shall be subdued unto him, then shall the Son also himself be subject unto him that put all things under him, that God may be all in all.
1. Corinththians 15:27-28
These verses speak clearly of a hierarchy, not of an equality of power or position and this is not only true for Jesus' resurrection, but rather – as the text above shows – it will also be true when Jesus returns and has overcome his enemies.
Furthermore, these two verses state three times, that God has put all things under Jesus – i.e. that Jesus did not inherently have this power.
God the Father and Jesus Christ his Son will give you grace, mercy and peace. Those blessings will be with us because we love the truth.
2. John 3
Grace be with you, mercy, and peace, from God the Father, and from the Lord Jesus Christ, the Son of the Father, in truth and love.
2. John 3
This is a listing of (and a differentiation between ) two parties) only one of which is called God.
For though there be that are called gods, whether in heaven or in earth, (as there be gods many, and lords many,) But to us there is but one God, the Father, of whom are all things, and we in him; and one Lord Jesus Christ, by whom are all things, and we by him.
1. Corinthians 8:5-6
This is a teaching that can not be misunderstood: Many are considered to be gods, but for all that have the same faith as Paul has there is only one God and this is: the Father.
Then he went back to Nazareth with them, and he obeyed them. But his mother kept all these things like a secret treasure in her heart. Jesus became wiser and stronger. He also became more and more pleasing to God and to people.
This rather reminds us what was said about the human being Samuel:
The boy Samuel continued to grow stronger. He also became more and more pleasing to the Lord and to people.
1. Samuel 2:26
It is also said about Jesus:
Though he were a Son, yet learned he obedience by the things which he suffered; And being made perfect, he became the author of eternal salvation unto all them that obey him;
The God proclaimed by the Bible can not grow any further or be made perfect, because he has been perfect from the beginning. Accordingly it is written for example:
Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning.
Jesus died on the cross, an innocent man, to carry the sin of other men. It would not make any sense for God to die for the sin of man.
Also if God died and knew he would rise again and did not suffer like a human, his death would have been a farce. This thought kept me from faith for a long time, since as a non-believer at that time I did not know that trinity is a human doctrine instead of being based on the Bible.
Death came because of what a man did. Rising from the dead also comes because of what a man did.
1. Korinther 15:21
God will bring Jesus back at a time that pleases him. God is the blessed and only Ruler. He is the greatest King of all. He is the most powerful Lord of all. God is the only one who can't die. He lives in light that no one can get close to. No one has seen him. No one can see him. Give honor and power to him forever.
1 Timothy 6:15-16
Which in his times he shall shew, who is the blessed and only Potentate, the King of kings, and Lord of lords; Who only hath immortality, dwelling in the light which no man can approach unto; whom no man hath seen, nor can see: to whom be honour and power everlasting.
1 Timothy 6:15-16
If God is immortal (cannot die), then Jesus could not have died either (if the Trinitarian doctrine was true). If nobody has died for us, then nobody could have saved us from death by dying in our place. Furthermore: If Jesus had not truly died, then he did not truly rise from the dead – and then we will also not truly rise from the dead.
The assertion, that Jesus is (also) God, triggers many further questions with regard to his death - inter alia:
Something else would have been a farce, if Jesus was God of equal substance and power: the three times Jesus was tempted by Satan in the dessert. Also Satan as a fallen angel would have known Jesus to be God and would not have tried to tempt him by promising him things that would have been Jesus' anyways. Furthermore Jesus did not say “I am God, forget it!”, rather he said “It is written” therefore forget it.
All this is so illogical that many seekers capable of thinking will be at least severely hindered by the trinitarian doctrine to accept faith in Christ.
Then Jesus was led by the Spirit into the desert to be tempted by the devil. After fasting for forty days and forty nights, he was hungry. The tempter came to him and said, If you are the Son of God, tell these stones to become bread. Jesus answered, It is written (Deuteronomy 8:3): 'Man does not live on bread alone, but on every word that comes from the mouth of God.' Then the devil took him to the holy city and had him stand on the highest point of the temple. If you are the Son of God, he said, throw yourself down. For it is written (Psalm 91:11-12): 'He will command his angels concerning you, and they will lift you up in their hands, so that you will not strike your foot against a stone.' Jesus answered him, It is also written (Deuteronomy 6:16): 'Do not put the Lord your God to the test.' Again, the devil took him to a very high mountain and showed him all the kingdoms of the world and their splendour. All this I will give you, he said, if you will bow down and worship me. Jesus said to him, Away from me, Satan! For it is written (Deuteronomy 6:13): 'Worship the Lord your God, and serve him only.'
(NIV - brackets added for easier reference)
God said to Mose who was exceptionally well acquainted with him:
But, he said, you cannot see my face, for no-one may see me and live.
Consequently it says about God, with the second coming of Jesus in mind
Which in his times he shall shew, who is the blessed and only Potentate, the King of kings, and Lord of lords; Who only hath immortality, dwelling in the light which no man can approach unto; whom no man hath seen, nor can see: to whom be honour and power everlasting. Amen.
1. Timothy 6:15-16
If Jesus truly had been God, men could not have looked at him, just as could not have looked at the one God, of whom the Old and the New Testament says, that nobody can look at him.
In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning.
This translation ignores a very important differentiation found in the original Greek text. There it says:
In the beginning was the Word, and the Word was with the God, and God was the Word. He was with the God in the beginning.
based on the interlinear-translation (please see text below)
Johannes 1, 1-2
Please look at these verses as they were in the original language:
If we read the text in the original language carefully, then it becomes apparent that the second mention of God (the one without the definite article) is likely to mean something different then in the other cases, possibly something like "divine" or "consistent with God". By not using the definite article the author would have said something about the quality of "word" (logos) instead of speaking about an identity (a second person) next to God. In doing so he could have said, that the power and wisdom of God is also reflected in his word. By contrast “the God” means the one and only God (as he was known from the Old Testament and as Jesus professed his Father to be – see John 17, 3 above). If no difference in meaning was intended, why would the article be used every time except for this one time, in conjunction with Word and God in these verses 1-2?
It becomes apparent, that one of the Bible quotes most often used to support the trinitarian doctrine does not lend much, if any, support at all, if we look a little more closely.
This argument is strengthened even further, by the fact that it does not say here “in the beginning was Jesus and Jesus was with God ...”. It talks about the “Logos” of God, his word, thoughts, concept … Only in verse 14 it says that the Logos became flesh.
So you must go and make disciples of all nations. Baptize them in the name of the Father and of the Son and of the Holy Spirit.
This is the second quote used to support the trinitarian doctrine. One quote alone if the whole rest of the Bible contradicts it should not be used as basis of a new doctrine. But I am certain this verse could also be read with a meaning that would be consistent with the rest of the Bible. If we bear in mind that the difference to the traditional Jewish faith are the inclusion of Jesus and the Holy Ghost in the profession of faith than the call for baptism can be read to mean to baptize in the faith of the three pillars of the same. This does not make them one or equal though!
Also, the phrase can not be assumed to be considered a teaching or instruction, since Acts tells only that the baptisms were made in the name of Jesus or in the name of the Lord etc (please check the following quotes).
Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.
Who, when they were come down, prayed for them, that they might receive the Holy Ghost: (For as yet he was fallen upon none of them: only they were baptized in the name of the Lord Jesus.) Then laid they their hands on them, and they received the Holy Ghost.
When they arrived there, they prayed that the believers would receive the Holy Spirit. The Holy Spirit had not yet come on any of them. They had only been baptized in the name of the Lord Jesus. Then Peter and John placed their hands on them. And they received the Holy Spirit.
While Peter yet spake these words, the Holy Ghost fell on all them which heard the word. And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost. For they heard them speak with tongues, and magnify God. Then answered Peter, Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we? And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.
When they heard this, they were baptized in the name of the Lord Jesus. And when Paul had laid his hands upon them, the Holy Ghost came on them;
Acts 19, 5-6a
Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life.
I and my Father are one.
John 10, 30
This can not to be read as one person or one substance They are indivisible as their goals and teachings are concerned, because Jesus is the Word of God. But this should be true for Christians too, as it was said by Jesus (and even if we would reach this goal we would still not be God):
Holy Father, keep through thine own name those whom thou hast given me, that they may be one, as we are.
That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us
that all of them may be one, Father, just as you are in me and I am in you. May they also be in us
I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me.
I will be in them, just as you are in me. I want them to be brought together perfectly as one. It will also show the world that you have loved those you gave me, just as you have loved me.
John 17, 23
You can read these verses in there context also in the Online-Bible.
Who is the image of the invisible God, the firstborn of every creature:
The first part of this verse seems at first glance to support the trinitarian view, but it is not, if we compare it with the account of creation:
So God created man in his own image, in the image of God created he him; male and female created he them.
We are all created in the image of God.  Because of sin Because of sin this image was distorted. Jesus however – without sin and the beginning of the new creation – is the undistorted image of God.
1 If there be therefore any consolation in Christ, if any comfort of love, if any fellowship of the Spirit, if any bowels and mercies, 2 Fulfil ye my joy, that ye be likeminded, having the same love, being of one accord, of one mind. 3 Let nothing be done through strife or vainglory; but in lowliness of mind let each esteem other better than themselves. 4 Look not every man on his own things, but every man also on the things of others. 5 Let this mind be in you, which was also in Christ Jesus: 6 Who, being in the form of God, thought it not robbery to be equal with God: 7 But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men: 8 And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross. 9 Wherefore God also hath highly exalted him, and given him a name which is above every name: 10 That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; 11 And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
1 Are you cheerful because you belong to Christ? Does his love comfort you? Is the Holy Spirit your companion? Has Christ been gentle and loving toward you? 2 Then make my joy complete by agreeing with each other. Have the same love. Be one in spirit and purpose. 3 Don't do anything only to get ahead. Don't do it because you are proud. Instead, be free of pride. Think of others as better than yourselves. 4 None of you should look out just for your own good. You should also look out for the good of others. 5 You should think in the same way Christ Jesus does. 6 In his very nature he was God. But he did not think that being equal with God was something he should hold on to. 7 Instead, he made himself nothing. He took on the very nature of a servant. He was made in human form. 8 He appeared as a man. He came down to the lowest level. He obeyed God completely, even though it led to his death. In fact, he died on a cross. 9 So God lifted him up to the highest place. He gave him the name that is above every name. 10 When the name of Jesus is spoken, everyone's knee will bow to worship him. Every knee in heaven and on earth and under the earth will bow to worship him. 11 Everyone's mouth will say that Jesus Christ is Lord. And God the Father will receive the glory.
If you take verse 6 or even verses 6 – 11 a in isolation, it would fit the trinitarian view of God. If you take in the context of the verses 1 – 11 however, the following becomes apparent
The Bible quotes cited here clearly indicate, that the doctrine of the holy trinity is in contradiction to the Bible and the words that Jesus said.
Jesus reprimanded the Pharisees and the other religious leaders for putting human teachings ahead of biblical truths (God's word). I am open to look at other verses of the Bible and would be grateful for being directed to relevant ones. But if I should not have overlooked something important, the trinitarian doctrine is one that leads men to take action that is wrong. In the least she diverts attention away from God, just like worshipping Mary (for other reasons) still does today Furthermore we contravene against the first commandment, if we put Jesus, against his own teaching, on one level with God.
Jesus wants us to follow him and love him, because he is the way to his father, his and our God, he has never encouraged him being worshipped like God. I have concentrated on showing that Jesus is not God, and in doing so shown indirectly what Jesus is.
It should be looked at the Holy Spirit and the doctrine of trinity in the same manner, but this would be beyond the scope of this presentation. In case of interest I will gladly oblige and prove that the Holy Spirit is not a separate person of a godhead.
The trinity doctrine is a dangerous, false doctrine, that keeps seekers from finding the answers in the faith in Jesus and his God and makes believers contravene against God's first commandment every time they for example sing songs that express trinitarian views.
"We worship one God in Trinity, and Trinity in Unity, neither confounding the persons nor dividing the substance. For there is one person of the Father, another of the Son, and another of the Holy Ghost. But the Godhead of the Father, Son and Holy Ghost is all one; the glory equal, the majesty co-eternal. Such as the Father is, such is the Son, and such is the Holy Ghost" excerpt from the Athanasian Creed (sixth century)
Tertullian (um 150 – um 230) still talked about three persons of one party (in a legal sense), but even at the Nicean Council in 325 Arius taught that Jesus was less than God, he was banned, but two years later reinstated,his teachings or variations thereof became dominant for the next few decades and were finally dropped at the second Nicean Council (381),
When I speak of major churches, I speak of large mainstream Christian churches. There are other large and small Christian churches that do not accept the Trinitarian view. Examples are the Mormons (Church of Christ of Latter Day Saints), the Jehovas Witnesses (which would both be viewed with some skepticism both by Catholics and Protestants), Christadelphians, some Messianic Jews (the letter two not having the extra texts that the former have).
There is nothing written about Mary after the scene under the cross and Pentecost quite contrary to that much is written about the disciples. Also her opinion is not referred to either, except for when she still tried to keep Jesus from following his path. Last but not least to Jesus the ties between those that do God's will were much more important than those between family members – and this specifically with regard to Mary and his siblings. (see Mark 3:32-35)
... for I, the LORD your God, am a jealous God, punishing the children for the sin of the fathers to the third and fourth generation of those who hate me, but showing love to a thousand generations of those who love me and keep my commandments.
Exodus 20: 5b
Even after the fall of mankind the fact of being created in or after the image of God gives humans their unique dignity:
Every moving thing that liveth shall be meat for you; even as the green herb have I given you all things. But flesh with the life thereof, which is the blood thereof, shall ye not eat. And surely your blood of your lives will I require; at the hand of every beast will I require it, and at the hand of man; at the hand of every man' brother will I require the life of man. Whoso sheddeth man' blood, by man shall his blood be shed: for in the image of God made he man.
Genesis 9: 3-6