For Washington, D.C. Part 1
Udvar-Hazy Center in Dulles, Virginia (the 'Temple of Aviation' among plane enthusiasts and far better than the National Air and Space Museum Mall Campus, but gets far less crowds):
American Art Museum (my favorite Smithsonian, but gets lost within the pack. Also suffers from not being on the Mall, though locals do frequent it):
Smithsonian American Art Museum by
angela n., on Flickr
National Building Museum:
Library of Congress - not really a hidden gem, but I've met dozens of tourists who go to the Capitol (used to do tours there as an Intern) but don't cross the underground walkway to the Library of Congress, which is just as - if not more - impressive
Arts and Industries Building - scuttlebutt is that it's the front-runner to become the new National Museum of the American Latino:
Arts of War and Arts of Peace, just behind the Lincoln Memorial
Blind Whino Arts Club
Eisenhower Executive Office Building, an enormous French-style structure that dwarfs the White House. Most tourists pass by on the outside, but the inside is also stunning:
Tea at Hillwood Estate. Gorgeous gardens, European luxury and rare faberge eggs.