The Trinity is One God with 3 Persons

Persons does not equal Gods.  It is a quality that an entity can have.

The idea of God in three persons is not new in the New Testament.

Genesis starts out with the statement that God using the word Elohim, which is plural, created the heavens and the earth.
Genesis 1:1
"In the beginning God (Elohim) created the heavens and the earth."
It mentions the Spirit of Gos in Verse 2:
"and the Spirit of God was hovering over the waters."
Then it says in verse 3:
"And God said, “Let there be light,” and there was light."
So even at Creation, we see God, The spirit of God, and the Word of God.

In verse Genesis 1:26, it says, “Let us make mankind in our image, in our likeness."
So there is a consciousness of a plurality of persons in the Godhead from the very first of creation in the first book of the Old Testament.

In Genesis 18, when God appears to Abraham, he appears as 3 men.
"The Lord appeared to Abraham near the great trees of Mamre while he was sitting at the entrance to his tent in the heat of the day. 2 Abraham looked up and saw three men standing nearby. When he saw them, he hurried from the entrance of his tent to meet them and bowed low to the ground.
3 He said, “If I have found favor in your eyes, my lord,[a] do not pass your servant by."

In Exodus 31, God, refered to as "The Lord," fills men with the Spirit of God to do good works.
"Then the Lord said to Moses, 2 “See, I have chosen Bezalel son of Uri, the son of Hur, of the tribe of Judah, 3 and I have filled him with the Spirit of God, with wisdom, with understanding, with knowledge and with all kinds of skills— 4 to make artistic designs for work in gold, silver and bronze, 5 to cut and set stones, to work in wood, and to engage in all kinds of crafts."
Notice that the Lord could have said "I filled him with my spirit," but instead says, "filled him with the Spirit of God."
This the Spirit of God is formalized, not as a part of God, but as a person of God.

In Psalm 110:1, David refers to a future descendent of his as "my Lord."
"The Lord says to my lord:
“Sit at my right hand
    until I make your enemies
    a footstool for your feet.”
God and the descendant of David are both refered to as "Lord."

The Messianic figure in the Old Testament is also given the qualities of God and worshipped.  A prime example is in Daniel 7:13.
"13 “In my vision at night I looked, and there before me was one like a son of man,[a] coming with the clouds of heaven. He approached the Ancient of Days and was led into his presence. 14 He was given authority, glory and sovereign power; all nations and peoples of every language worshiped him. His dominion is an everlasting dominion that will not pass away, and his kingdom is one that will never be destroyed."

It is also interesting to note that the Old Testament never says that God is a numerical one (Hebrew word 'yachid'), but rather a unity one (using the Hebrew word "echad").

Jesus taught that He and the Father are one and the same. One example is in John 14.
"6 Jesus answered, “I am the way and the truth and the life. No one comes to the Father except through me. 7 If you really know me, you will know[b] my Father as well. From now on, you do know him and have seen him.”
8 Philip said, “Lord, show us the Father and that will be enough for us.”
9 Jesus answered: “Don’t you know me, Philip, even after I have been among you such a long time? Anyone who has seen me has seen the Father. How can you say, ‘Show us the Father’? 10 Don’t you believe that I am in the Father, and that the Father is in me? The words I say to you I do not speak on my own authority. Rather, it is the Father, living in me, who is doing his work."

The interaction and unity of the Father, Son, and Holy Spirit is taught by John the Baptist in John 3:
"34 For the one whom God has sent speaks the words of God, for God[i] gives the Spirit without limit. 35 The Father loves the Son and has placed everything in his hands."

Jesus clearly unifies the Father, Son, and Holy Spirit in Matthew 28:19
"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,"
He gives one name for all 3.  That name would be YWHY - "I am that I am."

In John 15:26, Jesus sends the Advocate, who is the Spirit of Truth from the Father to the disciples.
"When the Advocate comes, whom I will send to you from the Father—the Spirit of truth who goes out from the Father—he will testify about me."
This shows that Jesus is equal to God to be able to do such a thing and that the Spirit is equal to Jesus in that he is able to do the work of Jesus.

Furthermore, the disciples taught the Trinity.  One good example is 1 Peter 1:2.
"who have been chosen according to the foreknowledge of God the Father, through the sanctifying work of the Spirit, to be obedient to Jesus Christ and sprinkled with his blood: Grace and peace be yours in abundance."

Conclusion: The Trinity is taught throughout the Bible and was taught by Jesus and his disciples as well.

Now, some people question the Trinity on the basis of their own thoughts.  God is not limited by the thoughts of men.  God is who he reveals himself to be.

Some may ask why then did Jesus say the Father is greater than him?

Many things can have such various qualities.

If I live in England, and I say "I'm going to the United States."  Then I say later, "I'm going to New York City."  Did I contradict myself?  

No, because New York City is in the United States.  If I go to New York City, I go to the United States.

Now Get this.  I can say the United States is Greater than New York City.  Yet, if I go to New York City, I go to The United States.

In the same way, Jesus can be God and say:

"My Father is greater than I" ( John 14:28)
or
"My Father is greater than all" (John 10:29)

Normal people could say they do things all on their own accord.  But, only if Jesus was God, could he be limited to doing only God's work as he said:
"I can of my own self do nothing" (John 5:30)