Washington DC has the Smithsonian, which is actually a number of museums. Monuments. And if you go out to Great Falls in nearby Virginia, the Blue Trail is a spectacular and somewhat rugged hike.
If in DC, make sure to take a tour of the Washington National Cathedral .. it wasn't built 600 years ago, but it looks like it with all he carvings, the soaring heights and the flying buttresses. It took decades to build and they employed some of the few remaining stonemasons in the world to carve the stones. From the top of the tower, you actually look DOWN on the Washington Monument (which is on lower ground).
The spot of Custer's last stand is pretty sobering. You stand at the top of the hill (where he made his last stand), the wind blowing in your hair and whipping the native grasses around you ... and look down at the tree-lined creek valley, imaging it full of teepees hidden by the trees and imagine how surprised he must have been at the sheer number of warriors.
But don't bother with Mount Rushmore ... it is a let-down.
Yellowstone National Park as well as the Grand Tetons are breathtaking. Lake Tahoe is gorgeous.
So are the Badlands of North Dakota, but don't camp near the streams - the mosquitos will swarm you.
New York City has a museum built from several small European castles, sitting on the cliffs overlooking the Hudson River - the Cloisters. Culture and museums are high on the list. Shopping for clothing too (if that is your thing).
All of the coast lines, both east and west, are great.
Go to Orange County, California for the beaches and the flea markets. Head north and do visit Disneyland. I was taken there as a guest, mainly because they wanted SO much to show me this stupid thing .. and it turned out I adored it ... as an adult who normally doesn't like that kind of stuff.
New Orleans, French Quarter, is full of character, jazz, and good entertainment. If you go, make sure to stop by the Cafe du Monde for a cup of coffee and some beignets fresh out of the deep fryer. Take plantation tours, bayou tours, ride the St. Charles streetcar through the old mansion district.
Hit the Newport Folk Festival in Rhode Island. We camped nearby and walking through the woods I stumbled across THE most beautiful country scene I have ever seen. It took my breath away. The festival was great too, but what stuck in my memory was that beautiful view over the river.