c.t.ner has a great list.
One little tidbit I would add is that you can sign up pretty much the day-of for tours of the Capitol. These tours are ok, but they are really limited and you are with a group.
The better way of seeing the Capitol is to go to one of the House or Senate office buildings (to the south - House, and north - Senate) and pop into any of the Congressional offices and ask for a gallery pass. Doesn't matter if you are not from their district, it is just a secretary at the front of the office handing them out.
Once you have a pass, you can actually see more of the Capitol and go watch either the House or the Senate in action. However, you need a separate pass for each... This leads me to my final tidbit on the topic.
Try going to office of D.C.'s delegate, Eleanor Holmes Norton. I believe her office has both the House and Senate passes because DC has no representative or senator. Her office is in the Rayburn House Office Building.
Another tip I have is to go up to the bar on the top of the W Hotel. They have a patio that has views of the entire mall - nice place to go.
Also. another free thing in DC is at the Kennedy Center. They have daily free performances at the Millennium Stage. More info at:
http://www.kennedy-center.org/programs/millennium/
Unfortunately, I am under impressed with the food in DC, so I can't really suggest anything special. People go to Hamilton's and things like that, but I find them over priced and mediocre. Food trucks are actually a pretty good option in DC.
I can't really say much about NOVA or Maryland, I try to stay out of the burbs as much as humanly possible... bridge and tunnel people, ugh...