The only two instructions the New Testament gives specifically to fathers are both negative.
Do not exasperate your children. Do not embitter them, or they will become discouraged. That is the whole of it — no list of duties, no description of headship in the house, nothing about provision beyond a general instruction to every relative. Whatever Paul thought fathers most needed telling, it was what to stop doing, and we should sit with that before we go looking for a job description. The eighteen passages below start where Scripture starts, which is not with human fathers at all but with the one they are all being measured against.
The picture everything else is measured against

Psalm 103:13
“As a father has compassion on his children, so the Lord has compassion on those who fear him.”
The comparison runs from the human to the divine, which assumes the reader knows what a good father looks like. For those of us who do not, the psalm is harder work than it looks.

Luke 15:20
“But while he was still a long way off, his father saw him and was filled with compassion for him; he ran to him.”
He saw him a long way off, which means he had been watching the road. Running was undignified for an older man in that world, and Jesus put it in for an audience who would have felt the embarrassment we no longer notice.

1 John 3:1
“See what great love the Father has lavished on us, that we should be called children of God! And that is what we are!”
John adds the second sentence as though he expects to be doubted. He is writing to people who had been told they did not belong, and he repeats himself because he expects us to need it twice.

Romans 8:15
“The Spirit you received does not make you slaves, so that you live in fear again; rather, the Spirit you received brought about your adoption to sonship. And by him we cry, ‘Abba, Father.’”
Abba is Aramaic and Paul leaves it untranslated in a letter to Romans, which tells us it had become a fixed word in the churches. It is ordinary family speech, and the point of it is that we are not being asked to be formal.
What Jesus taught men to call him

Matthew 6:9
“This, then, is how you should pray: ‘Our Father in heaven, hallowed be your name.’”
Our, not my, in a prayer he gave to individuals. The first word puts everybody praying it into the same household before anything is asked for.

Luke 11:11-13
“Which of you fathers, if your son asks for a fish, will give him a snake instead?… If you then, though you are evil, know how to give good gifts to your children, how much more will your Father in heaven give the Holy Spirit.”
Though you are evil is an extraordinary aside to drop into an encouragement. Jesus grants the flaw and argues from it anyway: even a bad father manages this much.

Malachi 2:10
“Have we not all one Father? Did not one God create us? Why do we profane the covenant of our ancestors by being unfaithful to one another?”
Malachi uses the fatherhood of God as an argument about how people treat each other. The doctrine arrives with an obligation attached rather than a comfort.
The only two instructions given to fathers

Ephesians 6:4
“Fathers, do not exasperate your children; instead, bring them up in the training and instruction of the Lord.”
Exasperate names a slow effect rather than a single act. Paul assumes a father can wear a child down by accumulation, without ever doing anything we would point to.

Colossians 3:21
“Fathers, do not embitter your children, or they will become discouraged.”
The result Paul fears is not rebellion but a child who has stopped trying. It is the quieter failure, and a house can carry it for years before anybody names it.
What a father is for

1 Timothy 5:8
“Anyone who does not provide for their relatives, and especially for their own household, has denied the faith and is worse than an unbeliever.”
Paul is writing about widows and who should be supporting them, so the provision he has in mind runs upwards as well as down. It is the harshest sentence he writes about a domestic failure.

2 Corinthians 12:14
“After all, children should not have to save up for their parents, but parents for their children.”
Paul says this to explain why he will not take money from a church he founded. The principle is offered as something so obvious it needs no defence, and it settles the direction money is meant to travel between generations.

Psalm 127:3-5
“Children are a heritage from the Lord… Like arrows in the hands of a warrior are children born in one’s youth.”
Arrows are made to be released, and their whole value is where they land. The image takes for granted that a father does not keep them.

Deuteronomy 6:6-7
“Impress them on your children. Talk about them when you sit at home and when you walk along the road, when you lie down and when you get up.”
Four times of day, which is a way of saying all of them. The teaching described has no slot in the week because it is not that kind of teaching.

Proverbs 20:7
“The righteous man walks in his integrity; his children are blessed after him.”
After him is the phrase. Proverbs is describing something that arrives later, in a life the father will not be watching. It is an unsatisfying reward, and it is the one we are offered.
Discipline, and how Scripture frames it

Proverbs 3:12
“Because the Lord disciplines those he loves, as a father the son he delights in.”
Delights in is the qualifier that governs the sentence. The discipline described here is inseparable from pleasure in the child, and any version without that is not what the verse is talking about.

Hebrews 12:7
“Endure hardship as discipline; God is treating you as his children. For what children are not disciplined by their father?”
Three verses later the writer concedes that human fathers disciplined us as they thought best. It is as close as Scripture comes to saying our own fathers were guessing, and it is said gently.

Proverbs 29:17
“Discipline your son, and he will give you peace; he will bring you the joy of your heart.”
The promised outcome is peace in the house and pleasure in the child. Proverbs measures the method by the state of the relationship afterwards, which is a test any of us can apply.
Fathers who were not the biological one

1 Corinthians 4:15
“Even if you had ten thousand guardians in Christ, you do not have many fathers, for in Christ Jesus I became your father through the gospel.”
Paul distinguishes a guardian from a father and claims the second word for himself. He had no children, and he is describing a relationship earned rather than inherited.

1 Thessalonians 2:11-12
“For you know that we dealt with each of you as a father deals with his own children, encouraging, comforting and urging you to live lives worthy of God.”
Three verbs, and all of them are things said rather than things done. Paul’s picture of fathering is mostly speech, and a few verses earlier he compares himself to a nursing mother as well.
What the job actually is
Less than the culture around the word suggests, and harder. Two negative instructions, a principle about which direction money moves, an image of arrows that assumes release, and a promise that the good of it shows up after the man is gone.
And a father is not required to be the standard. Scripture measures human fathers against God rather than the other way round, and when Jesus makes the comparison he allows in passing that the human end of it is evil and argues anyway. For anyone whose own father was absent or worse, that is the door: Paul claimed fatherhood over a church he had no blood relation to, and Jesus, dying, arranged a mother for a friend and a son for a woman who had just lost one.
