The Bible makes it plain that God is one but in three persons Father, Son and Spirit.
In Genesis 1:1-2 we read:
 
In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. (NIV)
 
This tells us that the Holy Spirit was involved in creation and therefore co-existent with God from the very beginning.
 
We read in John 1:1-3:
 
In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through him all things were made; without him nothing was made that has been made. (NIV)
 
Because 'Jesus' was the name given to the God to Son when he was born as a human baby, John uses the expression "The Word" (logos in Greek) to describe His previous eternal existence with God the Father. John explains that Jesus (the Son of God) was equally involved in the Creation.
 
We see the same in Hebrews 1:1-2:
 
In the past God spoke to our forefathers through the prophets at many times and in various ways, but in these last days he has spoken to us by his Son, ... through whom he made the universe. (NIV)
 
Jesus Himself claimed that He was God while He was on earth. See John 10:30: “I and the Father are one.” (NIV)

 
That claim sets Him apart from all other religious prophets or teachers. No other major religious prophet or teacher has actually claimed to be God like Jesus did! Later in John’s gospel we learn that that Jesus will send the Holy Spirit from the Father (John 15:26).
 
In Matthew's gospel we see Father Son and Spirit working together in Jesus’ baptism: as Jesus is baptised the Father speaks and the Holy Spirit descends on Him - see Matthew 3:16-17.
 
And the clearest reference to the trinity is given by Jesus Himself - when He sent His disciples out in the “name of the Father and of the Son and of the Holy Spirit” (see Matthew 28:19).
 
We know from James 1:17 that God does not change and so we believe that God has always existed in three persons from before the beginning of world history. That is why the belief that God is a trinity of three persons has been the settled doctrine of the Christian church since the very earliest days.
 
The exact detail of how God the Father, Son and Holy Spirit interrelate and co-exist is not easy for us to comprehend and is not clearly revealed in the Bible.
 
The early church produced a number of Creeds that help to give clarity to the doctrine of the Trinity. For example the Athanasian Creed that helps by putting together what the Bible reveals about the personhood of God.
 
Selected excerpts from the Athanasian Creed are reproduced below:
 

Athanasian Creed

...

3. … we worship one God in Trinity, and Trinity in Unity;
4. Neither confounding the persons nor dividing the substance.
5. For there is one person of the Father, another of the Son, and another of the Holy Spirit.
6. But the Godhead of the Father, of the Son, and of the Holy Spirit is all one, the glory equal, the majesty coeternal.
7. Such as the Father is, such is the Son, and such is the Holy Spirit.
8. The Father uncreated, the Son uncreated, and the Holy Spirit uncreated.
9. The Father incomprehensible, the Son incomprehensible, and the Holy Spirit incomprehensible.
10. The Father eternal, the Son eternal, and the Holy Spirit eternal.

…

13. So likewise the Father is almighty, the Son almighty, and the Holy Spirit almighty.
14. And yet they are not three almighties, but one almighty.
15. So the Father is God, the Son is God, and the Holy Spirit is God;
16. And yet they are not three Gods, but one God.
17. So likewise the Father is Lord, the Son Lord, and the Holy Spirit Lord;
18. And yet they are not three Lords but one Lord.

…

21. The Father is made of none, neither created nor begotten.
22. The Son is of the Father alone; not made nor created, but begotten.
23. The Holy Spirit is of the Father and of the Son; neither made, nor created, nor begotten, but proceeding.
24. So there is one Father, not three Fathers; one Son, not three Sons; one Holy Spirit, not three Holy Spirits.
25. And in this Trinity none is afore or after another; none is greater or less than another.
26. But the whole three persons are coeternal, and coequal.
27. So that in all things, as aforesaid, the Unity in Trinity and the Trinity in Unity is to be worshipped.
28. He therefore that will be saved must thus think of the Trinity.
29. Furthermore it is necessary to everlasting salvation that he also believe rightly the incarnation of our Lord Jesus Christ.
30. For the right faith is that we believe and confess that our Lord Jesus Christ, the Son of God, is God and man.
31. God of the substance of the Father, begotten before the worlds; and man of substance of His mother, born in the world.
32. Perfect God and perfect man, of a reasonable soul and human flesh subsisting.
33. Equal to the Father as touching His Godhead, and inferior to the Father as touching His manhood.
34. Who, although He is God and man, yet He is not two, but one Christ.