The mission of the Christian family is actually the mission for all families. Some choose to ignore God and let the family unfold out of happenstance. However, this is not the correct approach, and we, as Christian men, must consciously guide our families by the directives God gave us.
The Three-Fold Mission of the Family
This core, three-part mission is true for every family on earth, including non-Christian families. Deciding to ignore God and his commands does not grant permission to run a family any way we choose.
Part 1: Make More Christians
And these words, which I command thee this day, shall be in thine heart: And thou shalt teach them diligently unto thy children, and shalt talk of them when thou sittest in thine house, and when thou walkest by the way, and when thou liest down, and when thou risest up. - Deuteronomy 6:6-7 (KJV)
The first third of the mission for the family is in making more Christians. It is our duty to raise our children in the fear and admonition of the Lord. Teaching them through catechisms, family worship, and relating everyday life back to God and His truths.
Family worship is the most neglected aspect of the family duty to make more Christians. It's quite easy to let life take up our time in other activities less profitable. So, we must dedicate and protect time to family worship of our Lord.
This worship can be as short as reading a passage of scripture, singing a (theologically correct) hymn, and praying. For families with young children, that is all you'll be able to pull off, but as they grow, add in reflection questions to get the family thinking about the text.
Part 2: Create Culture & Economy
Train up a child in the way he should go: and when he is old, he will not depart from it. - Proverbs 22:6 (KJV)
The second mission of the family is to create culture and economy. It is through families that these two emerge. Yes, families can be influenced by the culture of other families, and some families even abdicate their responsibility to create culture to school, television, or the Internet. It's by this abdication that the man loses control of the family, its traditions, and ultimately its unique contribution to the Kingdom of our Lord Jesus Christ.
Just as culture is created by families, so is the economy. Each family has the duty of passing down skills and knowledge to future generations. As I'm sure it's obvious, this is mostly freely given over to schools. Families rely on schooling to give their children the necessary skills to develop the economy. I'm not saying you have to homeschool your children, but you do need to contribute heavily to the skills of your children.
Part 3: Assist the Head of the Family
And the Lord God said, It is not good that the man should be alone; I will make him a help meet for him. ... And the Lord God caused a deep sleep to fall upon Adam, and he slept: and he took one of his ribs, and closed up the flesh instead thereof; And the rib, which the Lord God had taken from man, made he a woman, and brought her unto the man. And Adam said, This is now bone of my bones, and flesh of my flesh; she shall be called Woman, because she was taken out of Man. - Genesis 2:18, 21-23 (KJV)
The final part of the threefold mission of the family is in assisting the head on his mission. Man was made first, and then God created woman, and by extension children, to help the man. The first man, Adam, had the mission of tending to the garden, and God saw he needed help on his mission.
Of course, many men do not have a mission, and therefore this part of the family mission cannot be filled. The first two parts are listed as such because they do not rely on this third point. It's best to get the first two in order and happening consistently before focusing on pulling the flock into the mission of the man.
When all three are exercised in unison, the family operates at a high level of fruitfulness. This fruitfulness will pour out into all the areas and lives that family touches. It creates a glimpse of what the pre-fall family would have been.