30 days? You could go by way of Seattle and San Diego if you really wanted to!
For scenic but not fast, taking some or all of Skyline Drive and the Blue Ridge Parkway is a nice option, particularly if you like to do a little bit (or a lot) of hiking. Do plan for the tunnels, some of which are a bit low, if needed. Most are towards the southern extreme of the parkway (which is also in my opinion the more scenic portion). The campgrounds actually on the road do not have hookups, but generally have dump stations and potable water fill stations.