So far I have only been speaking about the Son, as it is absolutely central that we see that he is truly God, like the Father. There is much less in the NT about the Holy Spirit, but that is because one of the main purposes of the Holy Spirit is to bring glory to Jesus. The Son is the central person in all of Scripture--much of the OT points to him and his sacrifice; in his name alone we have salvation. So it makes sense that we read less of the Holy Spirit.
Having said that, there is enough that we can see that the Holy Spirit is a distinct person from both the Father and the Son, yet is also truly God. For example:
Joh 14:16 And I will ask the Father, and he will give you
another Helper, to be with you forever,
Joh 14:17 even
the Spirit of truth, whom the world cannot receive, because it neither sees him nor knows him. You know him, for he dwells with you and will be in you. (ESV)
Joh 15:26 “But when
the Helper comes,
whom I will send to you from the Father,
the Spirit of truth,
who proceeds from the Father,
he will bear witness about me. (ESV)
First, the Holy Spirit is "another," that is, one who is similar to Jesus but distinct. That speaks of personhood. Second, "Helper" clearly speaks to personhood, especially since it also means "advocate." An advocate can only be a person. Here is how
parakletos is defined:
NAS Word Usage - Total: 5
- summoned, called to one's side, esp. called to one's aid
- one who pleads another's cause before a judge, a pleader, counsel for defense, legal assistant, an advocate
- one who pleads another's cause with one, an intercessor
- of Christ in his exaltation at God's right hand, pleading with God the Father for the pardon of our sins
- in the widest sense, a helper, succourer, aider, assistant
- of the Holy Spirit destined to take the place of Christ with the apostles (after his ascension to the Father), to lead them to a deeper knowledge of the gospel truth, and give them divine strength needed to enable them to undergo trials and persecutions on behalf of the divine kingdom
https://www.biblestudytools.com/lexicons/greek/nas/parakletos.html
John uses the same word here, for "advocate":
1Jn 2:1 My little children, I am writing these things to you so that you may not sin. But if anyone does sin, we have an
advocate with the Father, Jesus Christ the righteous. (ESV)
Notice that Jesus is said, by John, to be an advocate. It is not insignificant, then, that he records Jesus saying that he will send "another
parakletos."
Additionally, we see these acts of personal agency attributed to the Holy Spirit:
Acts: Matt 4:1; Acts 8:39, 16:7
Listens: John 16:13
Speaks: John 16:13-15; Acts 1:16, 10:19, 11:12, 11:28, 13:2, 15:28; 1 Tim 4:1; Heb 3:7
Can be lied to: Acts 5:3, which is the same as lying to God (5:9)
Bears witness: Rom 8:16; Heb 10:15; 1 John 5:6
Helps, intercedes, and searches: John 14:16, 15:26, 16:7; Rom 8:26-27; 1 Cor 2:10
Teaches: Luke 12:12; John 16:13; 1 Cor 2:13
Gives gifts: 1 Cor 12:11; Heb 2:4
Leads: John 16:13; Gal 5:18, Heb 9:8
Can be grieved: Eph 4:30
Can be outraged: Heb 10:29
Can be blasphemed: Matt 12:31-32
Convicts: John 16:8-11
And on it goes.
Of course, we also have the Great Commission:
Mat 28:19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, (ESV)
Notice that "name" is singular. This is what M. R. Vincent says in his Word Studies in the New Testament regarding this verse:
"The
name, as in the Lord's Prayer (“Hallowed be thy name”), is the expression of the sum total of the divine Being: not his
designation as God or Lord, but the formula in which all his attributes and characteristics are summed up. It is equivalent to
his person. The finite mind can deal with him only through his name; but his name is of no avail detached from his nature. When one is baptized into the name of the Trinity, he professes to acknowledge and appropriate God in all that he is and in all that he does for man. He recognizes and depends upon God the Father as his Creator and Preserver; receives Jesus Christ as his only Mediator and Redeemer, and his pattern of life; and confesses the Holy Spirit as his Sanctifier and Comforter."
I would also add that since we know the Father and the Son are distinct persons yet both truly God, it necessarily follows that the Holy Spirit is a distinct person that is also truly God. And since know there is only God, the Trinity is the best explanation.