Downtown DC -- the tourist part (National Mall, Smithsonian, monuments, and all) -- is great for biking. Nice and easy, great sites.
The Mount Vernon (paved) trail is 15 miles long and runs next to the river in Virginia down to (you see this coming, I'm sure!) Mount Vernon. I haven't ridden that trail, but people I know who have really like it.
Check out www.bikewashington.org for more ideas.