I've done both DC to the West Coast and DC to South Florida many times. Be advised that an instrument rating is very helpful, almost mandatory, for longer trips, particularly in the Southeast. Back when I was VFR-only, my trip postponement rate was about 50% in a DC summer. However, Great Plains to Southern California is usually VFR.
With a family of three, the economics are pretty good if you own the airplane. If you have a rental plane and get weathered in, costs will rise. IFR helps here.
Flight plan so the fuel stop has a good lunch spot on the field or near it. We found that a good supply of books, a drawing board, and a music source that feeds into the headset are key to keeping kids happy.
Oh, yes -- if you plan an overnight stay, make sure there is a pool that is usable and, ideally a restaurant at the hotel. Some stops are better than others -- my daughter, given a choice of Ponca City, OK, and Paris, France would go for Ponca every time. "It has everything you would want or need in a small area."
Long trips are actually better than short ones is some ways. If you are flying California to DC and don't like the weather forecast for Wyoming, the route through New Mexico is the same distance.
Paul