In short, the Holy Trinity is the belief that the one God subsists in three persons: God the Father, God the Son, Jesus (who took flesh in the incarnation and became man) and God the Holy Spirit. They are all God, with the same divine attributes, but relate to each other as Subject and Object.
It is ultimately a deep mystery, as we cannot fully understand how three can be one. It seems to go against logic. Yet the Bible teaches this clearly, with many and varied proofs, and therefore we must accept the revealed doctrine with faith, bowing to the fact that the thoughts of God are much higher than ours (Isa 55: 9). .
The Trinity is a classic case where there is little “direct” evidence, but a lot of deductive or indirect evidence, which can hardly be dismissed by anyone who accepts the inspiration of Holy Scripture: the revelation of God.
No passage says, “The one God exists in three persons: the Father, the Son, and the Holy Spirit.” Yet, for example, we see a verse that strongly suggests the same thing, with just a small inference:
Matthew 28:19 (KJV) Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,
If the Bible teaches that God (and God alone) has certain characteristics, and proceeds to apply them to three persons: called the Father, the Son, and the Holy Spirit, then they are all one God (since the Bible teaches that there is only one: De 6: 4; 32:39; Isa 43:10; 44: 8; 1 Cor 8: 4-6).
In my research, I found 40 passages that mention the three divine Persons. Here are eight (only one fifth):
Isaiah 61: 1 The Spirit of the Lord GOD is upon me,. . . (cf. 61, 2; Jesus applies this to Himself in Lk 4, 16-30)
Luke 3: 21-22 . . . when Jesus also had been baptized and was praying, heaven opened,  and the Holy Spirit descended on him in bodily form, like a dove, and a voice came from heaven: “You are my beloved Son; with you I am very happy. (cf. Mt 3, 13-17)
John 15:26 But when the Counselor comes, whom I send to you on behalf of the Father, the Spirit of truth, who proceeds from the Father, he will bear witness to me; (cf. 14:26)
Acts 2:33 Therefore being exalted at the right hand of God, and having received from the Father the promise of the Holy Spirit, he has poured out what you see and hear. (cf. 7:55)
Acts 20:28 Take heed to yourselves and to all the flock, in which the Holy Spirit has made you overseers, to take care of the Church of God which he obtained with the blood of his own Son.
Romans 3:30 p.m. I call you, brothers, through our Lord Jesus Christ and through the love of the Spirit, to struggle with me in your prayers to God on my behalf (cf. Eph 2:18)
1 Corinthians 6:11 . . . justified in the name of the Lord Jesus Christ and in the Spirit of our God. (cf. 1 Peter 1: 2)
2 Corinthians 13:14 The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all.
Is the Holy Spirit directly called God? Yes; here is the best single passage along those lines:
Acts 5: 3-4 But Peter said, “Anani’as, why has Satan filled your heart with lying to the Holy Spirit and withholding part of the income of the land?  . . . You did not lie to men but to God. . . .
Ananias lied to the Holy Spirit; at the same time he lied to God; therefore the Holy Spirit and God are synonymous: one and the same thing.
This sort of thing happens over and over again in the Bible: characteristics equivalent in many ways are applied to the three divine Persons:
1. Who raised Jesus from the dead? Well, that was God the Father (Ga 1: 1; 1 Thess 1:10); it was also Jesus himself (Jn 2,19; 10,17-18); and it was the Holy Spirit (Rom 8:11).
2. Who gave the new covenant? The Father (Jer 31: 33-34); Jesus (Heb 8: 1-13; 10:29; 12:24; 13:20); the Holy Spirit (Heb 10: 15-17).
3. Who sanctifies believers? The Father (1 Thess 5:23); Jesus (Heb 13:12); the Holy Spirit (1 Peter 1: 2).
4. Who is the creator? The Father (Gen 1: 1; Isa 44:24; Acts 17:24; Eph 3: 9); Jesus (Jn 1: 3; Col 1: 16; Heb 1: 8, 10); the Holy Spirit (Jb 33: 4).Job 33: 4.
5. Who inhabits the believers? The Father (1 Cor 3: 16a; 2 Co 6:16; 1 Jn 3:24); Jesus (Jn 6:56; Rom 8:10; Eph 3:17); the Holy Spirit (Jn 14: 16-17; Rom 8: 9, 11; 1 Cor 3: 16b). The Bible even describes this in terms of different combinations: Father and Son (Jn 14:23); Father and Holy Spirit (Eph 2: 21-22; 1 Jn 3:24); Son and Holy Spirit (Ga 4: 6).
What one Person does, others also do in perfect harmony and unity, and People “interpenetrate” one another. Christian theology has 50 cent words for this: circumcession (latin) or perichoresis (Greek).
Many things are very difficult to understand, yet firmly believed; to begin with: quantum mechanics, the physics of black holes, the nuclear fusion that occurs in the center of our sun, the “curvature” of space and time (Einstein’s relativity). The physical reality turned out to be very “bizarre” and unpredictable. The theology is also sometimes striking, and seemingly “strange” and unfathomable. This should not surprise us at all (since God is an extraordinary Being).
The cumulative arguments based on dozens of individual indications become very convincing: just like a large rope, made up of many individual strands woven together, is extremely strong. This is the nature of the biblical indications for the Holy Trinity. We find them at every turn. No one should go astray into thinking that the Holy Trinity is not “biblical”. Having seen the many reasonable proofs, we believe in the faith.