Cool post, River. If you swing back through the DC area (I'm not an active hobbyist at the moment, but did notice you recently visited), and if you wind up bringing your shiny new ride next time, check out BikeWashington.org. For any others interested, a few easy-to-access suggestions:
- C&O Canal Toe Path: 180+ miles from Georgetown to Cumberland MD. Gravel / dirt path that follows the C&O canal and the Potomac River. Nice short-ish ride is to Great Falls (about 15 miles one way).
- Mount Vernon Trail: mostly paved ride on the VA side of the Potomac. About 36 mile out and back from base of Key Bridge (Teddy Roosevelt Island) to Mt Vernon (nice ascent over the last 3.5 miles heading into the turn around).
- Rock Creek Park: easily accessible from a good portion of downtown DC. Road / trail gets a little hinky to follow about Connecticut Ave and the Zoo.
- Capital Crescent Trail: about a 20 mile out out and back (depending on where you pick it up) that goe from DC (Georgetown area) up to Bethesda and Silver Spring MD. Again, mostly paved, and some on-road sections.
I'm less familiar with the off-road options that you may be gunning for, but a neighbor swears by Prince William Forest (a bit of a drive outside the city though).
Have fun out there 
I know you're a fellow Chicagoan. I ride all the time too. Don't know if you knew it or not but there are trails being built along the Cal Sag that will be the longest trails in Chicago. You'll be able to get on them near Midway and go all the way to Joliet... Sections of them are already done and they're GREAT! Wooded and quiet, with few other riders, although I have been on weekdays so that helps.
I primarily do road biking in the DC area. In addition to the rides mentioned by pjones, I would suggest the outer section of the W&OD trail (from Rt. 28 to the west). If you want to see any monuments, I would suggest the trails around the Tidal Basin and Hains Point (provided you get there early enough to beat the crowds). In any case, I'd love to see you on your next trip to D.C. area.