Jesus’ teaching about the new covenant people of God in the Gospels:
1. How one becomes a member of new people of God.[I wonder if this kind of stuff would make better subject matter for a church membership class than some of the Baptist booklets that I've seen, read, and had re-read to me again and again or every time I moved and changed church and applied for membership].
2. How one is to live in relation to God, other members, and the wider world.
3. How one is to celebrate membership.
4. What one is to expect as God fulfils his purpose for and through his people.
The teachings about the church in the apostolic fathers and Didache.
1. The prospect of martyrdom for those committed to Jesus and the new community.[Maybe this stuff was emphasized in the apostolic fathers and didache, but I'm fairly sure that I could find these themes in the Gospels, Acts, Paul and Catholic epistles in varying degrees. Sounds alot like Hebrews to be honest.]
2. The need for obedience to leaders in the church.
3. How Christianity is to be differentiated from Judaism
4. How Christians are to behave within their own community and the wider world
5. What is true of Christian doctrine and what Christians should expect in the future.
In some, I like the first set of points, but I'm not so sure that the second set of points represents a distinct development in the post-apostolic period.
Howard Clark Kee, The Beginnings of Christianity: An Introduction to the New Testament (New York: T&T Clark/Continuum, 2005), 64-65.