The trinity is a doctrine that was established in the 2nd Century and was first given written form by Tertullian in his statement of the trinity: three beings (hypostases) in one substance (homoousios). The trinity is not consistent with Jewish belief, that of one God, and inconsistent with Jesus' own belief. Paul, also, did not write of a trinity. The early Christians in Jerusalem did not believe in a trinity. The first hints of a Trinitarian doctrine being formed are at the end of the 1st Century, as there are some hints in John. However, even that gospel is not Trinitarian.
The bible does not teach the concept of a triune God. The bible teaches that God is God, Jesus is the potential messiah who will bring about God's kingdom, God's power on Earth (the Holy Spirit) is within us for the asking.
Perhaps the most heavily used verses in support of the Trinity are John 1:1 and John 10:30.
John 1:1 has translated Logos as "Word" since Jerome's Latin Vulgate's use of "Verbum". However, this does not accurately translate the Greek. Logos is a complex construct in English giving the idea of: reason, meaning, dialogue. So, John 1:1 can as easily be translated as "In the beginning was reason", or "In the beginning was meaning". The attributing of a person to the "Word" is as bad as the Gnostics attributing a person to the "Wisdom" (Sophia) of God. Jesus is being described as God-sent, but not God. God's purpose (the establishment of God's kingdom) made flesh, i.e. the coming of the messiah.
In John 10:30 it states: I and my Father are one. However, could this phrasing not also mean “one” as in “of one mind”. In the Bible, Jesus also states that he and his disciples are “one” and that all nations will become “one”. Does this mean that Jesus and his disciples are one essence? Of course not. Is everything one essence? Well, perhaps – if one is inclined toward pantheism.
Another favourite is the "I am" in John 8. Jesus was not claiming to be God, he was claiming to be the messiah (the one sent by God) who Abraham foresaw the coming of.
There are many Biblical verses supporting Unity. John 8:17-18 states that there are two witnesses: Jesus and God (who sent him). If they were the same essence, there would only be the one witness – not valid in Jewish law. Jesus himself states that there is only one God (Matthew 19:17) and that God is greater than he (John 14:28). That Jesus was anointed by God (Acts 10:38) and given power by God (Matt. 28:18) to be the mediator between God and man (1 Tim. 2:5) and a doer of God's will (John 6:38). Paul wrote that Jesus interceded on our behalf, as an intermediary not as a direct appeal (1 Corinthians 8:6, Romans 16:27, Romans 7:25). In Hebrews, Jesus is described as being our High Priest.
In Matthew 19:17, Jesus asks why they call him good when there is but one who is good. The implication in this statement, especially when going on to the commandments in the next verse, is the one is God. Jesus is differentiating himself from God, as he does in many other verses (e.g. John 5:30, John 7:16-18, John 14:10, Mark 13:32). Jesus states that God works through him and that he does God's works. He doesn't state that he is God or equal to God.
In short, to support Unity one only has to show the Bible. In order to support Trinity, one has to understand an unfathomable mystery and accept it with faith. Why not just accept the mystery as unfathomable and learnt to accept that the path we're on is one we follow by ourselves. We can only learn through Jesus' example and hope to meet our Father along the way.