Much of what you find yourself thinking about when you start to look at what the Scripture teaches about God as Father, Son, and Holy Spirit is how the three persons in God relate to each other. There’s so much in scripture about it, even though it’s often not the first thing you notice when you’re reading. Personally, I’ve discovered that, for some reason, I had developed this way of sort of “skipping over” and not noticing the things in any given verse that give us insight into how the Father relates to the Son, and so on.
Along those lines, I found this summary of lots of Biblical data from Fred Sanders very helpful:
The Father loves the Son, sends the Son, and glorifies the Son. He also sends the Holy Spirit in Jesus’ name, in response to his request, and is worshiped in the Son and in the Spirit. He and the Son indwell one another. He has life in himself and has given to the Son to have life in himself. He is the judge and has committed judgment to his Son.
The Son was with God in the beginning, in the bosom of the Father, and was and is God. He made all things. He was sent by the Father, became flesh, and lived among men. He obeys the Father, prays to the Father, and after his resurrection ascends to the Father. He asks the Father to send the Holy Spirit, sends the Holy Spirit himself, and breathes out the Spirit on his disciples. He and the Father indwell one another. He receives from the Father life in himself and the right to judge.
The Holy Spirit proceeds from the Father, is sent by the Father on the Day of Pentecost in response to the Son’s request, and is also sent at that time by the Son. He is breathed out by the Son. He bears witness of the Son and brings glory to him.
The three work together in harmony. Through the Holy Spirit, they come together to the disciples, who as a consequence live in the Father and in the Son.
[From The Deep Things of God: How the Trinity Changes Everything]