The role of Father is built into Creation. All living creatures were created to be self-replicating; even the simplest single-cell creatures exchange genetic material and reproduce. The continued existence of every form of life on the planet is dependent on its ability to leave descendants. (Yes, not all of them do succeed, but that is part of the dynamics of constant change that we see.) The first human beings were created with the ability and responsibility to have children. It is interesting that people were not given the programming that most other creatures have; we were expected to teach our children. The other half of the story is that we also have the ability to choose our behavior. Hopefully, our parents instruction will enable us to make good choices. We find that Adam's record was't that good. Of the first two sons, one did it according to the instruction, and one didn't--and didn't seem to understand or pay attention, other than to take it out on his brother.
It is interesting that almost all references to God and fatherhood refer God as "the God of your fathers, Abraham, Isaac and Jacob." The only referrals to God as our Father that I could find are in Psalm 89:26 "He will call out to me, ‘You are my Father, my God, the Rock my Savior.’"; Isaiah 9:6 "For to us a child is born, to us a son is given, and the government will be on his shoulders. And he will be called Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace;" and Malachi 2:10 "Do we not all have one Father? Did not one God create us?" The coming of Jesus, prophesied in the Isaiah passage, refers to Jesus as the Son of God the Father. At the very end of Jesus' time on earth, we find Jesus saying, “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.’”
St Paul is very specific about this relationship; every single one of his letters, begins like this:" Grace and peace to you from God our Father and the Lord Jesus Christ." (I Corinthians 1:3) You could make an interesting study just from these verses--all slightly different in emphasis. Also, Peter, John, James and Jude say something similar as well. I especially noted this:"I keep asking that the God of our Lord Jesus Christ, the glorious Father, may give you the Spirit of wisdom and revelation, so that you may know him better" (Ephesians 1:17) Think about the role of father: one of the main jobs of a father is to teach; not only his children, but his community. Paul says here that the source of wisdom, revelation and knowledge come from God through His Spirit.
An important way that He communicates with his children is through prayer. Think for a moment in how we usually pray; and also how we communicate with our parents. Most of our prayer time is spent telling God what we want and what we think He wants to hear. Is this how we usually talk to our parents, grandparents, and others? If we want to learn from our parents, the best way is to listen to them. I recall when I was a kid out camping with my family, Dad would make a bonfire on the riverfront near where we were camped, and we'd sit around and Dad would tell stories of his life, and other family and friends and their stories. Dad loved to listen to people, as well--he would say that he'd never met someone he couldn't learn something from. He'd even ask me questions that come up in things he read. This makes me thing that prayer needs to be more like this--more listening than talking. It's not an easy thing to learn, we tend to be afraid of "dead time". I find one question, or observation, or even a tree to stare at, and then wait for a response. If something comes into your head, give to to the Spirit see what He does with it. Praise isn't meant to be flattery or an attempt to get something in return, but thankfulness and acknowledgment of who God is and who we are. This is God, the power behind everything that is, and He loves us and wants His best for us.