3-4 weeks is not long enough to come across country and back a different route.
I would stay off the Interstate highways and use federal US highways and major state routes.
Suggest not making reservations too far in advance. Normally you can find campsites with no reservations except at major parks (not RV) and holiday weekends.
Use a map to start with. Mark the states that you think you want to visit. Go to each states tourism site and order their vacation planner and a state map.
Start marking them up, then have each person mark them up as to what looks interesting to them.