Don't be afraid of the Metro - it will get you everywhere you need to go.
This.
Check out Chinatown (Green/Yellow line, Chinatown-Gallery Place Stop for some neat places...mainly the National Portrait Gallery).
Air and Space is on Independence between 4th and 7th SW.
Natural History and American History are also good stops. They're on Constitution on the other side of the Mall.
The Capitol visitors center is a nice place to spend some time in, but you'd be wasting your time if you don't take a tour. You can't just walk into the Capitol. Feel free to stop in your Congressperson's office and say hello. Google it.
Roosevelt Island is a nice place few tourists know about. Take the Orange Line to Rosslyn in Arlingon, VA and hike a bit. It's worth it. Also, you could check out Georgetown from the same Metro stop. There's a shuttle.
Dupont Circle has a couple of good places as well. On the Red Line.
And located just a couple hundred feet away from my home, Meridian Hill Park on 16th Street. A medium size walk from the Columbia Heights stop on the Green/Yellow line.
Old Town Alexandria on the Yellow/Blue line (King Street stop) is a safe bet.
But, if you're restricting yourself to just whats around GAO, you're going to limit yourself. GAO is in what's called the "Southwest Federal Center" and there isn't much there after hours...and during hours it's secure federal offices. NASA is there and has a gift shop if you're in the mood for mission pins, but everything shuts down after 5pm. Luckily the L'Enfant Metro station sits on the Blue/Orange and the Green/Yellow line. You could get to wherever you want with minimal hassle.
