Trinity - God in three persons

JN 5:31 "If I testify about myself, my testimony is not valid. There is another who testifies in my favor, and I know that his testimony about me is valid."
 
Jesus here says that for his testimony to be valid in a court of law (his claims are on trial here in this chapter), he would need another person to testify with him.
 
The other person is the Father (John 5:37). Two persons are required here, for Jesus' testimony to be valid in the legal sense that Jesus is claiming here.
 
And this concept appears in other aspects in Scripture as well, to paint for us in miniature what the Trinity means in the Godhead:
 
MT 19:5-6 "For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh." So they are no longer two, but one.
 
"No longer two"! And yet though there is "one flesh," there are still two people:
 
EPH 5:31 "For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh." This is a profound mystery--but I am talking about Christ and the church.
 
And this same concept also applies in our relationship with Christ, the church, all of God's people, are united with him in this way, yet we are still many people, not one person:
 
1CO 12:27 Now you are the body of Christ, and each one of you is a part of it.
 
The marriage relationship is a picture in miniature of the Trinity, as is the church in Christ: One body of Christ, but more than one person, one flesh, but two people.
 
Other examples are a woman with child:
 
The child is inside the woman, they are connected physically, one body, in a sense, and the woman is in a "mother" relationship to the child, yet they are both distinct persons, yet sharing all the potential of a human being, especially all the potential of a human spirit.
 
And the cherubim of Ezekiel:
 
Ezekiel 1:5-6  In appearance their form was that of a man, but each of them had four faces and four wings.
And singular and plural switch back and forth in this description, in verse 10, it's "their faces," in verse 15, it's literally "his faces," in verse 19, it's "the living creatures," in verse 20, it's (literally) "the living creature." And lest we think that only the man-aspect-face is conscious and sentient, we have another description in this verse, apparently the faces were not even static:
 
Ezekiel 10:14 And each one had four faces. The first face was the face of a cherub, the second face was the face of a man, the third the face of a lion, and the fourth the face of an eagle.
 
And both cherubs and men are certainly sentient. Now people might say this is only metaphorical, but even if it is, this is a clear picture of multiple beings, yet one being, in Scripture.

As far as "Elohim" (a plural form name for God) being a real plural, there are verses where a plural verb form is used with "Elohim":
 
Genesis 20:13: "And it came to pass, when God (Elohim) caused me to wander [literally: They caused me to wander] from my father's house...
 
And yet in a verse with a plural verb used with Elohim, we also have singular pronouns, "your," "his" and "himself" etc. are singular, as is "a name," though "went" is plural:
 
2 Samuel 7:23 And who is like your people Israel, the one nation on earth whom God went to redeem to be his people, making himself a name and doing for them great and awesome things by driving out before your people, whom you redeemed for yourself from Egypt, a nation and its gods?
 
And here El is used with Elohim and a plural verb, again indicating both plural and singular:
 
Genesis 35:7 and there he built an altar and called the place El-Bethel, because there God had revealed himself to him when he fled from his brother.

"Hear, O Israel, Adonai Eloheinu Adonai is one. These three are one. How can the three Names be one? Only through the perception of faith; in the vision of the Holy Spirit, in the beholding of the hidden eye alone.…So it is with the mystery of the threefold Divine manifestations designated by Adonai Eloheinu Adonai—three modes which yet form one unity."(Zohar II:43b, vol. 3, p. 134 in the Soncino Press edition))
This text speaks of a three-fold God, though in terms of modes, but this is not from the New Testament and is definitely not by a Christian author, but from a Jewish perspective, from the Zohar.

Jesus was worshiped, and will be worshiped, with no rebuke:
 
Matthew 2:11 On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him.
 
Now we cannot read "worshiped him" as "bowed down" here, for that would be "they bowed down and bowed down." And similarly here:
 
Matthew 28:9 Suddenly Jesus met them. "Greetings," he said. They came to him, clasped his feet and worshiped him.
Matthew 14:33  Then those who were in the boat worshiped him, saying, "Truly you are the Son of God."
Matthew 28:17 When they saw him, they worshiped him; but some doubted.
Luke 24:52 Then they worshiped him and returned to Jerusalem with great joy.
John 9:38 Then the man said, "Lord, I believe," and he worshiped him.
Revelation 5:12-14 In a loud voice they sang: "Worthy is the Lamb, who was slain, to receive power and wealth and wisdom and strength and honor and glory and praise!" Then I heard every creature in heaven and on earth and under the earth and on the sea, and all that is in them, singing: "To him who sits on the throne and to the Lamb be praise and honor and glory and power, for ever and ever!" The four living creatures said, "Amen," and the elders fell down and worshiped.
 
So Jesus is really a member of the Godhead, and one of the persons of the Godhead.
 
Also, the Holy Spirit is a person, and a member of the Godhead:
 
First are the several verses where the Holy Spirit is put in a coordinate relationship with the Father and the Son (Matt. 28:19; 1 Cor. 12:4-6; 2 Cor. 13:14; Eph. 4:4-6; 1 Peter 1:2): since the Father and Son are both persons, the coordinate expression strongly intimates that the Holy Spirit is a person also. Then there are places where the masculine pronoun "he" is applied to the Holy Spirit (John 14:26; 15:26; 16:13-14), which one would not expect from the rules of Greek grammar. Moreover, the name counselor or comforter is a term commonly used to speak of a person who helps or gives comfort or counsel to another person or persons, but is used of the Holy Spirit in John's gospel (14:16,26; 15:26; 16:7).
 
Other personal activities are ascribed to the Holy Spirit, such as teaching (John 15:26), bearing witness (John 15:26; Rom. 8:16), interceding or praying on behalf of others (Rom. 8:26-27) ...
 
Finally, if the Holy Spirit is understood simply to be the power of God, rather than a distinct person, then a number of passages would simply not make sense, because in them the Holy Spirit and his power or the power of God are both mentioned. For example, Luke 4:15, "Jesus returned in the power of the Spirit into Galilee," would have to mean, "Jesus returned in the power of the power of God into Galilee." (Wayne Grudem, "Systematic Theology")

"Allow me now, most earnestly, to impress upon you the absolute necessity of being sound on the doctrine of the Trinity. I knew a man, a good minister of Jesus Christ he is now, and I believe he was before he turned aside unto heresy — he began to doubt the glorious divinity of our blessed Lord, and for years did he preach the heterodox doctrine, until one day he happened to hear a very eccentric old minister preaching from the text, 'Thy tacklings are loosed: they could not well strengthen their mast, they could not spread the sail.' 'Now,' said the old minister, 'you give up the Trinity, and your tacklings are loosed, you cannot strengthen your masts. Once give up the doctrine of three persons, and your tacklings are all gone, your mast, which ought to be a support to your vessel, is a rickety one, and shakes.' A gospel without the Trinity! — it is a rope of sand that cannot hold together. But, give me a gospel with the Trinity, and the might of hell cannot prevail against it; no man can any more overthrow it, than a bubble could split a rock, or a feather break in halves a mountain. Get the thought of the three persons, and you have the marrow of all divinity. Only know the Father, and know the Son, and know the Holy Ghost to be One, and all will appear clear. This is the golden key to the secrets of nature; this is the clue of the mazes of mystery, and he who understands this, will soon understand as much as mortals ever can know.' (C.H. Spurgeon)