The best climbing gym is probably the Earth Treks in Rockville, MD. It's HUGE and Earth Treks is a very good organization. If you live on the Virginia side of the Potomac, I hear SportRock is very good but I've never been there.
The best trad climbing for weekend trips is either Seneca Rocks or The Gunks. You're probably about three hours from Seneca and about five from The Gunks.
For day trips, Old Rag is supposed to have some great routes but I've never been there. The approach is a bit off putting for a lot of people. Maryland Heights has some cool routes and a very nice view of Harper's Ferry. It's also one of the few places in Maryland for multi-pitch. Rocks State Park, north of Baltimore, also has some good stuff worthy of an occasional day trip. Annapolis rocks has excellent single pitch trad but the approach puts some people off, again.
There are probably some other nice areas besides Old Rag on the Virginia side but I don't know them because I'm based in Baltimore.
Earthtreks in Rockville md is definitely the premier gym here. Has decent, lead-able cracks too (clipping bolts along crack). Take it from a former Berkeley-living, Yosemite/Tahoe climbing excile who moved to DC 6 months ago for the same reason; you are giving up a bit on the climbing side, but DC is a cool place worthy of spending some time, particular if the job is fulfilling. Give a shout when you get out here. I'd be up forflitting the gym or weekend cragging anytime.
Sportrock in Alexandria is a friendly place, the one out in Sterling seems to be mainly family oriented with tons of kids. I have not been to earthtreks but hear only good things about it. There's no local crags, sorry. The gunks doable for weekend trips, but you'll have to cross new jersey and nyc on friday night and there's a lot of traffic. The new river is about the same distance, 5-6 hours. It's a very interesting city but not a good place for climbing.
While not quite of typical California/Sierra quality, one should not neglect mentioning DC's backyard crags of Carderock and Great Falls. Sure they are tiny,made of weird, polished stone, and usually top-roped, but they are very accessible, historic, have a good number of interesting climbs of all grades---a good number of which are leadable (especially at Great Falls)and offer a great opportunity to get ones hands on actual rock in a pleasant outdoor setting within a very short drive of downtown (at least away from rush hour).