Don`t do a cross-country trip. You can get a favorable microcosm of the whole country by staying in California. Nature-wise it has all of the beaches, mountains, forests, deserts, etc. you could want, including snowy areas. The people in Southern California are different enough from the North so that you`ll have a different enviornment to observe. Go to the east area, away from the coast to get that kind of Middle-America feel.
LA would be difficult to navigate without a car because it is sprawly, but it`s worth it.