Definitely take advantage of the fact that all the museums and the zoo are free. Few major cities with this level of amenities just let you walk in and pay nothing. The art galleries on the mall are very good if you're into that sort of thing. The zoo is reasonably good if that's appealing as well.

Rock Creek Park is a pretty uniquely rustic park for being in the middle of a city. Good hiking/biking paths. You can use capital bike share easily enough to take a scenic ride. Especially down Beach, which the southern half of was completely closed off to cars. The zoo can actually be got to off of Rock Creek Trail very easily.

If you like urban bike baths, the Metropolitan Branch Trail is also very good. Goes from Union station up to neighborhoods that aren't often seen by tourists. You can hike this too, of course. On Saturdays, there's a brewery on this path that has good BBQ set-up outside.

I second the Air and Space museum that's way out in Virginia, near the Dulles airport. It's kind of a pain to get to and you have to pay for parking, but you get to see some seriously historical craft including the space shuttle Discovery.

Get some Ethiopian and El Salvadorian food while you're in town. Sadly, most of the good ethnic food in the area is in the suburbs, but those two have great representation in the city and are a bit unique as well.