Come up north, to Massachusetts. Some of the best seafood around, and you'll also discover why drivers from around here are universally referred to as 'Massholes' )))))))))))))
witness the gentrification!
it might be one of the better ways to get a sense of culture if you can catch a bohemian sort of area, before they turn into BMW dealerships.
west to east, as far as destinations are concerned i guess off the top of my head,
san fran's china town, the golden gate, seattle + space needle :3
grand canyon, mt. rushmore, yosemite & yellow stone park, decayed detroit, deep dish pizza in chicago, new orleans, driving through vermont, niagra falls / great lakes
i was taking high school physics when i visited kennedy space center, that was a really cool trip that tied things together steeped in the history and the narrative.
personally i would try to find people willing to let you stay in their home, you could be extreme and sort of couch surf, it seems renting places are trendy at the moment, but it's always nice to get a guide in places you think look cool, who better than people who live there. on the topic of accomodations too, since schools "out" if you're looking for places to stay, university dorms may be offered to rent out for a night, they'll usually be cheaper than hotels AND be situated in cool parts of town.
a tip i've learned while traveling is: follow the food!! you'll learn a lot about the area and the people that live there if you can find out where all the him places to eat are, where people are congregating, sharing a sandwich and a story. these will usually be the sort of epicenters of communities, and also give you a great feel for the place, and what they revolve around. it will be tough on the road, but if you plan a route, try and find any seasonal events that are going on, like BBQ cook outs in memphis for instance.