Oh, hell's bells, I ride on the sidewalk on the Mall up to the Capitol! I wouldn't ride on a sidewalk north of Constitution Avenue, but on the south sidewalk of Constitution and further south of it, I do it whenever I'm biking in DC, as do loads of other bikers.
(Good to know that it's not legal, though; now I'll smile when I ride!)
Another nice ride is the Capital Crescent Trail, which parallels the C&O for a bit and runs up to Bethesda. If you are going to ride the C&O, you want fat tires if you're going to head up to Great Falls (which is very pretty and good for nature). The C&O Canal towpath is rocky in the Great Falls area.
Rock Creek Park is good for nature, too. From Alexandria, you have easy access to Capital Crescent, C&O, and Rock Creek. Not so much for nature, but another nice little ride, if you're riding in DC, is around Hains Point. It's a different perspective on DC, and is a very very (very) popular loop for bikers, runners, skaters, etc.
(I was biking around Hains Point yesterday, and had to really work to pass one skater.... me passing someone almost never happens. Of course, she later passed me. I was doing what for me is a very speedy 14 MPH, and she zoomed right on by.)
Parts of Rock Creek are closed to motor vehicle traffic from 7 AM Saturday to 7 PM Sunday (and on holidays). The hiking is good in the park, too.