No, God is not a Trinity. The Trinity is described like this:
There is one God in Three persons,-the Father, the son, and the Holy Ghost. These three persons are co-equal in all things. The Father is God, the Son is God, and the Holy Ghost is God, and yet they are not three Gods, but one God.1
According to the doctrine of the Trinity, both Jesus and the Holy Spirit are God as well as the Father. However, the Bible says the only God is the Father:
…there is one God, the Father, from whom are all things and for whom we exist, and one Lord, Jesus Christ… (1Cor. 8:6)
There is a definite distinction between God, who is the Father, and Jesus. Interestingly, the Holy Spirit isn’t even mentioned — this is because the Holy Spirit is God’s mind (1Cor. 2:8-16) or power (Luke 1:35; 24:49; Acts 2:4), not a third person in the Godhead.
1Tim. 2:5 also makes the distinction clear:
…there is one God, and there is one mediator between God and men, the man Christ Jesus
Jesus is a distinct being from God (notice how he is also pointedly described as being a ‘man’, even after he ascended to heaven. And, again, notice the absence of the Holy Spirit: for the New Testament writers there was no thought of the Holy Spirit being a member of the Godhead).
John 17:1-3 is also very clear on this point. In John 17 Jesus is praying to his Father (v1), and in v3 he describes his Father as ‘the only true God’ (and he describes himself as the person whom the only God has sent — i.e. he’s a different being to God):
{1} When Jesus had spoken these words, he lifted up his eyes to heaven, and said, “Father, the hour has come; glorify your Son that the Son may glorify you, {2} since you have given him authority over all flesh, to give eternal life to all whom you have given him. {3} And this is eternal life, that they know you the only true God, and Jesus Christ whom you have sent.
The Father is God, and Jesus is the person whom he has sent (the Holy Spirit isn’t even mentioned as being in this relationship). Jesus and the Holy Spirit are not God.
Also see John 3:16; 14:28; 20:17; 1Tim. 6:15-16 (NB: no one has seen God, but lots of people saw Jesus); 1Peter 1:3; Jude 1:25; Rev. 1:1. These all show a distinction between God and Jesus, showing that Jesus is not God. For occasions when Jesus is called God, see the answers to Why does Isaiah 9:6 call Jesus “Mighty God, Everlasting Father”? and What did Thomas mean when he said to Jesus “My Lord and my God”? (John 20:28).
It is important to know who God is, and who Jesus is, because, as we have seen in John 17:3, knowing God, and knowing Jesus, whom he sent, is ‘eternal life’. God wants us to worship in truth (John 4:24). We need to know who God is so that we can truly follow the Lord Jesus words — and his example — and ‘love the Lord your God with all your heart and with all your soul and with all your mind’ (Mat. 22:37).
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Footnotes
1. V. Staley, The Catholic Religion: A manual of Instruction for Members of the Anglical Church, (Mowbray), p. 152