The Washington Area Bicycling Association (WABA) has lots of information about riding in DC. There's also goDCgo and Capital BikeShare (good resources for cycling for transportation in the DC area, which is the best way to get around).

I took wonderful spin classes at a rec center in Fairfax, but it's pretty far from Old Town. Not sure about classes in or near Old Town. But with all the great cycling in and around DC, who needs spin classes, at least in the spring, summer, and fall??

There are quite a few DC-area TEers here, and I'm sure they will chime in. Welcome to the area and enjoy your riding.