(NASB) But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things, and bring to your remembrance all that I said to you.
Jesus Himself speaks of our baptism into His passion and death, so as to be raised with Him (Galatians 3:27; Romans 6:3):
Matthew 28:19 (NASB) Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit
God has a Son, and a Spirit, speaking in human terms. Intricately connected and considered together.
The Holy Spirit is He that proceeds from the Father, God, (as to His generation, or origin as a divine person), and testifies of the Son (John 16:14). He has life from the Father via the Son (again, intimate union), just as the Son Himself lives because of the Father (John 16:15; 6:57).
The same sanctifies us and applies the merits of Christ to us (1 Corinthiains 6:11), convicts the world of its sin (John 16:8).
John the Baptist says, with reference to Jesus, when Jesus was about to begin His ministry and preach the good news of His redemption of mankind that:
Matthew 3:11 (NASB) As for me, I baptize you with water for repentance, but He who is coming after me is mightier than I, and I am not fit to remove His sandals; He will baptize you with the Holy Spirit and fire.
When Jesus appeared to his disciples alive again, encouraging them to preach the good news to the whole world, He makes reference to this:
Acts 1:4-5 (NASB) Gathering them together, He commanded them not to leave Jerusalem, but to wait for what the Father had promised, "Which," He said, "you heard of from Me; for John baptized with water, but you will be baptized with the Holy Spirit not many days from now."
Then we read that on the day of Pentecost, 40 days after Christ rose from the dead, the following happened:
Acts 1:14-16, 2:22 (NASB) These all with one mind were continually devoting themselves to prayer, along with the women, and Mary the mother of Jesus, and with His brothers. At this time Peter stood up in the midst of the brethren (a gathering of about one hundred and twenty persons was there together), and said, "Brethren, the Scripture had to be fulfilled, which the Holy Spirit foretold by the mouth of David concerning Judas, who became a guide to those who arrested Jesus.
Notice also that the Holy Spirit is the One who inspires the writers of Scripture. The Apostle Peter speaks of Him when he writes: (ASV) For no prophecy ever came by the will of man: but men spake from God, being moved by the Holy Spirit.
When the day of Pentecost had come, they were all together in one place. And suddenly there came from heaven a noise like a violent rushing wind, and it filled the whole house where they were sitting. And there appeared to them tongues as of fire distributing themselves, and they rested on each one of them. And they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit was giving them utterance. Now there were Jews living in Jerusalem, devout men from every nation under heaven. And when this sound occurred, the crowd came together, and were bewildered because each one of them was hearing them speak in his own language.
Because the Holy Spirit of God is God's Spirit, He is called God:
Acts 5:3-4 (NASB) But Peter said, "Ananias, why has Satan filled your heart to lie to the Holy Spirit and to keep back some of the price of the land? "While it remained unsold, did it not remain your own? And after it was sold, was it not under your control? Why is it that you have conceived this deed in your heart? You have not lied to men but to God."