Cherry hill is a good option, though not exactly inexpensive (but that's not to say overpriced; they offer a lot and charge correspondingly).
Not far from them there's Greenbelt National Park, which is not expensive but has very few ammenities (no hookups, typical rustic public campground) but lots of sites and a reasonable walk of maybe a mile and a half to the nearest Metro station.
East of the city there's Lake Farifax county park, which offers intermediate ammenities (hookups, etc.) and likewise is not too distant from a Metro station. I think all three campgrounds are around a mile or two walking from their respective nearest Metro stations, or a bit farther by car on the roads.
As bgum says, you'll be there a very long time to begin to see everything.