A lot of interesting places in my interesting guidebook aren't open anymore, but the Octagon House is. I don't know why it's called that since it isn't actually an octagon, but it's a good place to visit for lots of reasons. There's official American history- the Treaty of Ghent was signed there because the Madisons lived there after the White House was burned. There's architecture- it's a federal-style building that is now owned by the American Institute of Architects. There's family history- the Tayloe family lived here for years. There's lots of other history too, especially in the 50-year gap between the time the Tayloes moved out and the AIA took over. And it's even haunted.
And they let you wander around as you please. You can download several mp3 tours, or they have a nice brochure that tells about all the different rooms.