DC resident here. Lots of good things in this thread. Couple of additions.

First, I don't think a car is nearly as much of a hassle as is made out. Parking is expensive, and can be a challenge on weekdays. But other than rush hour, it's much easier driving around the city than out in the suburbs!

I would add that the Nationals and DC United play at RFK stadium in the city. Great Falls is great for rock climbing/hiking. I prefer MD side. Canal/towpath are pretty interesting for a bike ride, with some working locks. You can rent a paddle boat by the Jefferson memorial, do some sailing on the Potomac (short drive down the GW parkway in VA), or go to Annapolis/Baltmore Inner Harbor for nautical stuff.

There's an IMAX in Air and Space and in the museum right across the Mall (Natural history I think). The Awakening statue at the very tip of Hains Point is pretty cool, and there's a public golf course, public tennis courts, and pool there as well. Kennedy Center has great (free) afternoon/evening concerts in the public spaces, sometimes even music 13 year olds like! ESPN Zone is downtown for light entertainment (and drinks for you), and there's a new yuppy bowling alley next to MCI Center. Going up the Washington Monument at night is pretty wild. The Smithsonian Folklife Festival should be going on the Mall then.

Out of town, the ocean is 2.5 hours, Williamsburg/Kings Dominion about 2 hours, Baltimore and Annapolis slightly less than an hour. Mount Vernon is about half an hour. Blue Ridge mountains/sky line drive are less than 2 hours.