Just a sample of what Scripture teaches:
The Father and the Son are not the same person:
Matthew 11:27
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.
Galatians 4:4-7
4 But when the set time had fully come, God sent his Son, born of a woman, born under the law, 5 to redeem those under the law, that we might receive adoption to sonship.[a] 6 Because you are his sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, “Abba, Father.” 7 So you are no longer a slave, but God’s child; and since you are his child, God has made you also an heir.
John 17:3
3 Now this is eternal life: that they know you, the only true God, and Jesus Christ, whom you have sent.
Ephesians 1:17
17 I keep asking that the God of our Lord Jesus Christ, the glorious Father, may give you the Spirit[a] of wisdom and revelation, so that you may know him better.
John 8:16
16 But if I do judge, my decisions are true, because I am not alone. I stand with the Father, who sent me.
John 20:17
17 Jesus said, “Do not hold on to me, for I have not yet ascended to the Father. Go instead to my brothers and tell them, ‘I am ascending to my Father and your Father, to my God and your God.’”
Matthew 27:45–46
45 From noon until three in the afternoon darkness came over all the land. 46 About three in the afternoon Jesus cried out in a loud voice, “Eli, Eli,[a] lema sabachthani?” (which means “My God, my God, why have you forsaken me?”).
1 Thessalonians 1:10
10 and to wait for his Son from heaven, whom he raised from the dead—Jesus, who rescues us from the coming wrath.
The Son and the Spirit are not the same person:
Acts 10:38
38 how God anointed Jesus of Nazareth with the Holy Spirit and power, and how he went around doing good and healing all who were under the power of the devil, because God was with him.
Romans 8:11
11 And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies because of[a] his Spirit who lives in you.
The Father and the Spirit are not the same.
John 14:26
26 But the Advocate, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you.
John 15:26
26 “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.
Father is God:
Philippians 1:2
2 Grace and peace to you from God our Father and the Lord Jesus Christ.
Jesus is God:
John 20:28
28 Thomas said to him, “My Lord and my God!”
John 1:1
In the beginning was the Word, and the Word was with God, and the Word was God.
Colossians 2:9
9 For in Christ all the fullness of the Deity lives in bodily form,
Holy Spirit is God:
Acts 5:3–4
3 Then Peter said, “Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? 4 Didn’t it belong to you before it was sold? And after it was sold, wasn’t the money at your disposal? What made you think of doing such a thing? You have not lied just to human beings but to God.”
Plurality from the beginning:
Genesis 1:26
26 Then God said, “Let us make mankind in our image, in our likeness, so that they may rule over the fish in the sea and the birds in the sky, over the livestock and all the wild animals,[a] and over all the creatures that move along the ground.”
Some of the Roles of the Father:
Draws people to the Son (John 6:44)
Father gives sheep to the Son (John 10:29)
Father has a Son (Psalm 2:7, 12)
Ultimate source/cause of universe (1 Corinthians 8:6; Revelation 4:11)
Divine revelation (Revelation 1:1)
Salvation (John 3:16-17)
Jesus' human works (John 5:17; 14:10)
Chooses who will be saved (Eph 1:4)
Some of the roles of the Son:
Atones for Sin (1 John 2:2)
Creation and maintenance of the universe (1 Corinthians 8:6; John 1:3; Colossians 1:16-17)
Divine revelation (John 1:1; 16:12-15; Matthew 11:27; Revelation 1:1)
Salvation (2 Corinthians 5:19; Matthew 1:21; John 4:42)
Redeems (Eph 1:7)
Holy Spirit:
Seals the believer's Salvation (Ephesians 1:13)
Creation and maintenance of the universe (Genesis 1:2; Job 26:13; Psalm 104:30)
Divine revelation (John 16:12-15; Ephesians 3:5; 2 Peter 1:21)
Salvation (John 3:6; Titus 3:5; 1 Peter 1:2)
Jesus' works (Isaiah 61:1; Acts 10:38)
Holy Spirit seals (Eph 1:13)
Other verses dealing with Personhood:
Distinction between Father and Son (John 14:23)
Son sent by the Father (John 6:38)
Father gives sheep to the Son (John 10:29)
Father is Greater than Son (John 14:28)
Paul says “God and Lord Jesus Christ” (2 Thessalonians 1:2, 12)
Jesus is the image the invisible God (Colossians 1:15)
David says “The Lord says to My Lord(Psalm 110:1)
The Lord Has a Son (Psalm 2:7,12 Proverbs 30:2-4)
Spirit distinguished from Lord (Numbers 27:18)
Spirit distinguished from God ((Psalm 51:10-12)
God the Son is distinguished from God the Father (Psalm 45:6-7; Hebrews 1:8-9)
Says the Father will send a Helper, the Holy Spirit (John 14:16-17)
Son makes reference to Father and Spirit ( Isaiah 48:16 and 61:1)
Compare Isaiah 61:1 to Luke 4:14-19 to see that it is the Son speaking.
Spirit descents upon Son, father is pleased (Matthew 3:16-17)
Baptize in name of each person (Matthew 28:19)
Grace of Jesus, Love of God, fellowship of Holy Spirit (2 Corinthians 13:14)
As was mentioned already, you have a pretty well irrefutable text demonstrating the distinctiveness of the three persons.
Matthew 3:13-17
13 Then Jesus came from Galilee to the Jordan to John, to be baptized by him. 14 John would have prevented him, saying, “I need to be baptized by you, and do you come to me?” 15 But Jesus answered him, “Let it be so now, for thus it is fitting for us to fulfill all righteousness.” Then he consented. 16 And when Jesus was baptized, immediately he went up from the water, and behold, the heavens were opened to him, and he saw the Spirit of God descending like a dove and coming to rest on him; 17 and behold, a voice from heaven said, “This is my beloved Son,[c] with whom I am well pleased.”
Conclusion? The doctrine of the Trinity is true.
