I recently bought a bike -- after a 10-year stint without one! -- and have had a blast exploring Northern Virginia's myriad bicycle and jogging trails.
The Washington, D.C., area has miles and miles of trails for bikers, joggers, rollerbladers, etc., and I've only just scratched the surface.
You'll want to take the Mt. Vernon Trail, which follows the Potomac River from Rosslyn (in Arlington) all the way down to Mt. Vernon, George Washington's home, about 20 miles away. Along the way, you'll get great views of our nation's capitol, including of the Kennedy Center, the National Mall, and some of the beautiful bridges that span the river.
Be sure to stop off by National Airport to watch the planes take off and land right above your head!
Visit Arlington County's Web page (http://www.arlingtonva.us/Departments/EnvironmentalServices/dot/planning/bike/EnvironmentalServicesBicycling.aspx) for trail maps and the rules/laws that govern them. Note that any rider under the age of 14 MUST have a helmet.