Jefferson Memorial and Natural History Museum are the best monument and museum IMO.
Be sure to check out the clock tower of the Old Post Office. It's free, and is one of the best views in D.C.
Georgetown has lots of nice shops and houses and it's fun to watch the snobs.
Capitol Hill has some really cool rowhouses. Be sure to check out the market at Eastern Market on Saturday.
Old Town Alexandria is very nice, like Georgetown but much more laid-back.
Good, reasonably priced food is everywhere. Check washingtonpost.com for reviews of places. If you are near the Mall and want something decent and cheap, hit up Union Station and the Ronald Reagan building food court.
Black Cat and 9:30 Club are cool for concerts. The Kennedy Center Millenium Stage has a different free concert every day. If you're out to drink, hit Adams Morgan or Georgetown.
Dupont Circle is cool to hang out in if you don't mind seeing people with "alternative lifestyles".
The Sculpture Garden on the Mall has an ice skating rink in the fountain during the winter.
Gravely Point is a great place to park and watch the planes. They take off and land right over you. Actually, this is proably the coolest place in D.C.
Annapolis and Batlimore's Inner Harbor are nice day trips.
Take Metro everywhere - get yourself and unlimited use pass. And don't worry about getting shot - handguns are illegal in D.C.
