Where are you going to be living? The closest airports to DC itself are Potomac (VKX), Hyde (W32), and College Park (CGS). If you want to fly out of there, though, you will need to get cleared by pretty much the full alphabet soup of agencies. You'll have to get printed at DCA, and the FBI/TSA will do a criminal background check. Then you'll have to go the FSDO for an interview. And then after that, they will clear you to fly.
I've flown out of both VKX and W32. I'm not sure if anything is happening at W32 anymore. The FBO there is called Beacon Flying Service. At VKX, they'll try to get you into their flight club, where you pay $350 a month for "unlimited" flight time, however, you're capped at 50 hours per six months. And then you have to pay for gas.
You could also check out Manassas (HEF) or Montgomery County (GAI). I fly out of GAI now, and it's a nice place. It's an easy to get to airport, and you don't have to deal with all the FRZ clearance BS.