What are ways in which God reveals Himself to a Christian in the context of a two-way, one-on-one relationship?

To respond to the question:

God reveals Himself through a manifestation.

A truly saved Christian does not need to see a Theophany (visible manifestation of the invisible God) in order to receive such a proper knowledge/manifestation of God.

Atheists, agnostics, and all those who don’t know God through Jesus Christ (including us before we are saved) know God exists (eternally) via a manifestation of Himself that is more general, namely through creation:

“For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of people who suppress the truth by their unrighteousness,

because what can be known about God is plain to them, because God has made it plain to them.

For since the creation of the world his invisible attributes – his eternal power and divine nature – have been clearly seen, because they are understood through what has been made. So people are without excuse.” ‭‭Romans‬ ‭1:18-20‬ ‭

This is a knowledge of God that God convey’s and communicates to all of mankind through His creation.

The truly saved Christian knows God through a direct (as opposed to) a general knowledge of God.

Scripture makes it clear that God manifests Himself to the Christian directly, and then indwells them; thus they have a more vivid knowledge of the truth:

“All things have been handed over to me by my Father. No one knows the Son except the Father, and no one knows the Father except the Son and anyone to whom the Son decides to reveal him.” ‭‭Matthew‬ ‭11:27‬ ‭

This epistemic environment of the heart and mind is filled with the Holy Spirit, who is God, we read this in 2 major places in scripture:

“And what mutual agreement does the temple of God have with idols? For we are the temple of the living God, just as God said, “I will live in them and will walk among them, and I will be their God, and they will be my people.” ‭‭2 Corinthians‬ ‭6:16‬

& again:

“Do you not know that you are God’s temple and that God’s Spirit lives in you?” ‭‭1 Corinthians‬ ‭3:16‬

This same Holy Spirit, who is our God, resides within God’s people (true Christians), to give them an internal understanding that is foreign to the unbelieving & agnostic community.

Hebrews 6:4-5 shows that after a believer “shares” or “partakes” of the Holy Spirit, they experience the following:

1.) “enlightened”

2.) “taste the heavenly gift”

3.) “taste the good word of God” (probably a reference to the scriptures)

4.) “taste of the powers of the coming age”

Such an experience is what makes the Christian know that the Bible is the Word of God. The Holy Spirit bears witness to the Word of God in writing via “tasting the good word of God”.

This is different than God manifesting Himself to the “World” of unbelievers in the sense that they don’t share in the illuminating power and presence of the Holy Spirit.

Jesus said the world doesn’t know the Holy Spirit:

“the Spirit of truth, whom the world cannot accept, because it does not see him or know him. But you know him, because he resides with you and will be in you.” ‭‭John‬ ‭14:17‬

‭Judas even asked about how Jesus will reveal Himself to the disciples and not to the world:

“The person who has my commandments and obeys them is the one who loves me. The one who loves me will be loved by my Father, and I will love him and will reveal myself to him.”

“Lord,” Judas (not Judas Iscariot) said, “what has happened that you are going to reveal yourself to us and not to the world?” Jesus replied,

“If anyone loves me, he will obey my word, and my Father will love him, and we will come to him and take up residence with him.” ‭‭John‬ ‭14:21-23‬ ‭

Cognitive knowledge doesn’t equate to special knowledge of God.

It is God who grants belief in Him, and “eyes” to see: (Philippians 1:29, Acts 26:15-18)