The word "Trinity" is never used in the Bible. But references to "The Father, Son, and Holy Spirit" are everywhere.

From the very first verse of the Bible:

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.

John places Jesus there in the beginning referring to him as "The Word of God":

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. In him was life, and that life was the light of all mankind. The light shines in the darkness, and the darkness has not overcome it... ...The true light that gives light to everyone was coming into the world. He was in the world, and though the world was made through him, the world did not recognize him. He came to that which was his own, but his own did not receive him. Yet to all who did receive him, to those who believed in his name, he gave the right to become children of God— children born not of natural descent, nor of human decision or a husband's will, but born of God. The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the one and only Son, who came from the Father, full of grace and truth.

References to prophets being "filled with the Holy Spirit" occur throughout the Old and New Testaments.

In Isaiah 9 we encounter the Father and Son as the same Being:

For to us a child is born, to us a son is given, and the government will be on his shoulders. And he will be called Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace.

In Matthew 3, we encounter all three in the same scene:

As soon as Jesus was baptized, he went up out of the water. At that moment heaven was opened, and he saw the Spirit of God descending like a dove and alighting on him. And a voice from heaven said, "This is my Son, whom I love; with him I am well pleased."

Jesus' last words bundled them together in one Name:

Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.

Jesus referred to the Holy Spirit as "The Helper" or "The Comforter" or "The Advocate"

However, I'm telling you the truth. It's for your advantage that I'm going away, because if I don't go away, the Helper won't come to you. But if I go, I will send him to you.

And Jesus explained the Spirit to one of the great Pharisees (while clearly expecting him to already know all about it):

Flesh gives birth to flesh, but the Spirit gives birth to spirit. You should not be surprised at my saying, 'You must be born again.' The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit." "How can this be?" Nicodemus asked. "You are Israel's teacher," said Jesus, "and do you not understand these things?

Finally, the Apostles' Creed, one of the oldest summaries of Christian beliefs embraced by Catholics and Protestants alike reads as follows:

• I believe in God the Father, Almighty, Maker of heaven and earth:
• And in Jesus Christ, his only begotten Son, our Lord:
  -- Who was conceived by the Holy Spirit, born of the Virgin Mary:
  -- Suffered under Pontius Pilate; was crucified, dead and buried:
  -- He descended into hell:
  -- The third day he rose again from the dead:
  -- He ascended into heaven, and sits at the right hand of God the Father Almighty:
  -- From thence he shall come to judge the quick and the dead:
• I believe in the Holy Spirit:
• I believe in the holy universal church:
  -- the communion of saints:
  -- The forgiveness of sins:
  -- The resurrection of the body:
  -- And the life everlasting.