Let me begin by first saying that the church is God's "program" for evangelism and discipleship. To be a Christian is to be a member of the Lord's church and the word "church" by original definition means "official assembly" or "congregation". So if you are part of the CHURCH you assemble locally with a church. I say that briefly to say that discipleship at home or worship as a family does not replace the accountability and fellowship a believer should commit to in a local church.
Now to the home, the Bible is clear that parents- particularly fathers- have a responsibility to bring the gospel to bear in their children's lives (not to mention each spouse's responsibility to do the same toward each other). Paul' s commands to the church in Ephesians 6:1-4 and Colossians 3:18-21 particularly remind us that the Christian home should exemplify the gospel. Sunday School teachers and youth ministers may provide help and accountability to come alongside the parents but are not a replacement for Christian parents sharing Scripture with their children.
I know this can be intimidating to some parents but if you treasure God's Word and love your children you will make this a matter of importance and for improvement. Keep it simple for starters. Have a set and protected time when your family prays together. Set apart time to read Scripture together. My eleven-month old has little understanding of what we're saying but before bed we huddle up and I pray for our family. When I read Scripture to him I realize he isn't getting much from it yet but it is great practice and discipline for me! As your kids grow (or even if its just you and your wife) discuss the sermon from Sunday or read the sermon text for the upcoming Sunday. Think about ways to "put it into practice". Remember above all the principle of Isaiah 55:11- God's Word will not return empty!
Here are some helpful links:
Children's Bible Reading Plan (David Murray)
Is your Youth Group accomplishing anything? (Dustin Nickerson/ Resurgence Blog)
How to Provoke your Children (John MacArthur)
The Centrality of the Home (sermon by Voddie Baucham)
The Family: Together in God's Presence (John Piper on families worshipping together)