Trinity is just a word invented as a shortcut to explain what the discussion/arguement is about. Others have tried to update the word a Triunity. There is no way to define an infinite God with the finite words of man. It all comes down to what the Bible claims.
The Father is God
We do not see God called as Father until quite late in History. Isaiah (740-687 BC) is the first to use the term in this way. Before that He is always referred to as the God of our Fathers.
For to us a child is born, to us a son is given; and the government will be upon his shoulder, and his name will be called “Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace.” Is. 9:6
He shall cry to me, ‘Thou art my Father, my God, and the Rock of my salvation.’ Psa. 89:26
No one has ever seen God; the only Son, who is in the bosom of the Father, he has made him known. John 1:18
Messiah’s claims
This was why the Jews sought all the more to kill him, because he not only broke the sabbath but also called God his Father, making himself equal with God. John 5:18
You do what your father did.” They said to him, “We were not born of fornication; we have one Father, even God.” Jesus said to them, “If God were your Father, you would love me, for I proceeded and came forth from God; I came not of my own accord, but he sent me. John 8:41-42
They are Separate
Jesus said to her, “Do not hold me, for I have not yet ascended to the Father; but go to my brethren and say to them, I am ascending to my Father and your Father, to my God and your God.” John 20:17
The Son is God
God said to Moses, “I AM WHO I AM.” And he said, “Say this to the people of Israel, ‘I AM has sent me to you.’” Ex. 3:14
Jesus said to them, “Truly, truly, I say to you, before Abraham was, I am.” John 8:58
4 Then Jesus, knowing all that was to befall him, came forward and said to them, “Whom do you seek?” 5 They answered him, “Jesus of Nazareth.” Jesus said to them, “I am.” Judas, who betrayed him, was standing with them. 6 When he said to them, “I am,” they drew back and fell to the ground. 7 Again he asked them, “Whom do you seek?” And they said, “Jesus of Nazareth.” 8 Jesus answered, “I told you that I am; so, if you seek me, let these men go.” John 18:4-8
The Holy Spirit is God
The earth was without form and void, and darkness was upon the face of the deep; and the Spirit of God was moving over the face of the waters. Gen. 1:2
Then the Spirit of the LORD came upon Jephthah, and he passed through Gilead and Manasseh, and passed on to Mizpah of Gilead, and from Mizpah of Gilead he passed on to the Ammonites. Judg. 11:29
Then Samuel took the horn of oil, and anointed him in the midst of his brothers; and the Spirit of the LORD came mightily upon David from that day forward. And Samuel rose up, and went to Ramah. 1Sam. 16:13
When thou sendest forth thy Spirit, they are created; and thou renewest the face of the ground. Psa. 104:30
Behold my servant, whom I uphold, my chosen, in whom my soul delights; I have put my Spirit upon him, he will bring forth justice to the nations. Is. 42:1
They made their hearts like adamant lest they should hear the law and the words which the LORD of hosts had sent by his Spirit through the former prophets. Therefore great wrath came from the LORD of hosts. Zech. 7:12
. . . and the Holy Spirit descended upon him in bodily form, as a dove, and a voice came from heaven, “Thou art my beloved Son; with thee I am well pleased.” Luke 3:22
And when they had come opposite Mysia, they attempted to go into Bithynia, but the Spirit of Jesus did not allow them; Acts 16:7
But you are not in the flesh, you are in the Spirit, if in fact the Spirit of God dwells in you. Any one who does not have the Spirit of Christ does not belong to him. Rom. 8:9
For the kingdom of God is not food and drink but righteousness and peace and joy in the Holy Spirit; Rom. 14:17
Do you not know that you are God’s temple and that God’s Spirit dwells in you? 1Cor. 3:16
There is only ONE God
“Hear, O Israel: The LORD our God is one LORD; Deut. 6:4
Not sure exactly how that works, but it is what the Bible teaches. I can explain Zeus, Odin and many other pagan gods, because they are the creations of men. Their petty lives and desires are just in the likeness of men. However the One True God is so beyond our finite minds ability to explain, no one in church history has ever done a complete job of it.
The Song of Songs 