What is the Trinity? Are the Father, Son, and Holy Spirit the same or different beings? What is their relationship with each other? Let's systematically address these questions.
Some say, "Yes" because Jesus said they are one:
John 10:30
I and the Father are one."
and…
John 14:9b
Anyone who has seen me has seen the Father. How can you say, 'Show us the Father'?
14:10 Don't you believe that I am in the Father, and that the Father is in me? The words I say to you are not just my own. Rather, it is the Father, living in me, who is doing his work.
14:11 Believe me when I say that I am in the Father and the Father is in me;
Jesus did not say, “I am the Father.”
John 17:11
I will remain in the world no longer, but they are still in the world, and I am coming to you. Holy Father, protect them by the power of your name—the name you gave me—so that they may be one as we are one.
and…
John 17:22
I have given them the glory that you gave me, that they may be one as we are one:
This does not mean Christians are supposed to be the same being.
Matthew 20:23
He said to them, "My cup you shall drink; but to sit on My right and on My left, this is not Mine to give, but it is for those for whom it has been prepared by My Father."
The Father acts, decides, and exists independently of Jesus.
Matthew 26:39
And He went a little beyond them, and fell on His face and prayed, saying, "My Father, if it is possible, let this cup pass from Me; yet not as I will, but as You will."
If they're the same, to whom was Jesus praying?
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,
We're to be baptized in the name of three separate entities.
Matthew 11:27
"All things have been committed to me by my Father. No one knows the Son except the Father, and no one knows the Father except the Son and those to whom the Son chooses to reveal him.
Jesus says He reveals the Father only to some.
Jesus says He's not the one who makes all the decisions.
Matthew 20:23
Jesus said to them, "You will indeed drink from my cup, but to sit at my right or left is not for me to grant. These places belong to those for whom they have been prepared by my Father."
The Father makes this decision.
Mark 13:32
"No one knows about that day or hour, not even the angels in heaven, nor the Son, but only the Father.
Jesus says He doesn't know this, but the Father does.
John 6:46
No one has seen the Father except the one who is from God; only he has seen the Father.
Jesus clearly indicates He's not the Father in saying He alone has seen the Father.
John 14:9
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'?
Here, Jesus is essentially saying there's nothing more to see. In order for humans to see the Father, He'd have to take on some physical form. This isn't significantly different from the form Jesus took. Since the Father and the Son are united in spirit (though not the same) there is nothing more to see.
Paul tells us Jesus is distinct from God, the Father.
1 Corinthians 8:6
yet for us there is but one God, the Father, from whom all things came and for whom we live; and there is but one Lord, Jesus Christ, through whom all things came and through whom we live.
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,
There are three into which we're to be baptized.
John 14:16-17
16 And I will ask the Father, and he will give you another Counselor to be with you forever--
17 the Spirit of truth. The world cannot accept him, because it neither sees him nor knows him. But you know him, for he lives with you and will be in you.
The Holy Spirit lives in us apart from Jesus.
John 16:7-10
7 "But I tell you the truth, it is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I go, I will send Him to you.
8 "And He, when He comes, will convict the world concerning sin and righteousness and judgment;
9 concerning sin, because they do not believe in Me;
10 and concerning righteousness, because I go to the Father and you no longer see Me;
Jesus had to leave the earth before the Holy Spirit could come.
Here, Jesus tells us blasphemy is handled differently by the Holy Spirit than it is for the Father and Son. This certainly indicates they are distinct entities.
Matthew 12:32
Anyone who speaks a word against the Son of Man will be forgiven, but anyone who speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come.
Mark 3:29
But whoever blasphemes against the Holy Spirit will never be forgiven; he is guilty of an eternal sin."
Luke 12:10
And everyone who speaks a word against the Son of Man will be forgiven, but anyone who blasphemes against the Holy Spirit will not be forgiven.
Therefore, based on these scriptures, they're not the same being. If Jesus had to go away for the Holy Spirit to come, they can't be the same being. If blasphemy against Jesus is treated differently from blasphemy against the Holy Spirit, they can't be the same being.
Paul clearly indicates the Father and the Son are different beings.
1 Corinthians 15:27-28
27 For he "has put everything under his feet." Now when it says that "everything" has been put under him, it is clear that this does not include God himself, who put everything under Christ.
28 When he has done this, then the Son himself will be made subject to him who put everything under him, so that God may be all in all.
Philippians 2:6
Who, being in very nature God, did not consider equality with God something to be grasped,
2:7 but made himself nothing, taking the very nature of a servant, being made in human likeness.
Here, Paul tells us just how Jesus is God—in nature, yet he indicates there is someone higher with which to be compared. Jesus is a separate being, Deity in nature, yet subordinate to the Father.
John 1:1-4
[1] In the beginning was the Word, and the Word was with God, and the Word was God. [2] He was in the beginning with God. [3] All things came into being through Him, and apart from Him nothing came into being that has come into being. [4] In Him was life, and the life was the Light of men.
John tells us Jesus was with God and that Jesus is God. Jesus was with God in the beginning, and all things came into being through Jesus.
Romans 8:9
You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you.
There you have the Trinity. All deity, all “God,” yet they are separate persons: One Father, One Son, One Spirit. This isn't a mystery. This is not even a difficult concept. They are one in purpose, love, and intent.