Off the top of my head --
There is much cycling for transportation in DC. Automobile traffic in the city is utterly insane. It literally makes me cry. If you're traveling within the city and you can do so by bike, you'll probably be much happier. There are lots of relatively new bike lanes and Capital Bikeshare is growing by leaps and bounds.
For group rides, I highly recommend the Oxon Hill Club. You simply will not meet a nicer group of people. They mostly ride in southern Maryland, which is the area that is east and south of DC. Many of their rides start within a 30-40 minute drive of where I live, in Arlington, so they would probably be about the same distance for someone coming from the city. There are some nice rural roads in southern Maryland. http://www.ohbike.org/
The largest club in the area is Potomac Pedalers. I like some of their rides and some of their members are friendly, but they also seem to have plenty of people who just want to ride as fast as they can with no interest in riding with the group or being the least bit friendly. The main benefit with PPTC is that they offer rides in different parts of the area, including Virginia and northern Maryland. www.potomacpedalers.org
Both clubs offer rides at various levels (in terms of average speed), though the vibe with the faster PPTC rides is survival of the fittest -- if you get dropped or have a mechanical, you're on your own.
There are some other clubs in the area, but I'm not as familiar with them. The Reston bike club (in Virginia) has a pretty limited ride schedule. I've considered joining the Baltimore Bike Club, though they start many of their rides more than an hour from where I live. They put on the Civil War century every year, which is one of my favorite rides. There's also Babes on Bikes, but I've never ridden with them because I think they mostly ride on weekdays.
I can't help you with racing, though I know there are various races and teams in the area. I do know a couple of women in the Oxon Hill club who race. My LBS has a team that does some racing, including cyclocross (I think).