Recent college grad myself — I grew up in the District and have returned home for the summer. I have spent my high school and college years running in the area and can offer a few tips.
As for D.C. itself, I would second Rock Creek Park, which has Beach Drive (paved winding road that is closed off on weekends) and quite a few trails to explore. RCP is great for long runs on weekends. These last few years, I have routinely done 15-18 miles by running down through the creek and back home through residential neighborhoods. There is also the Glover Archbold Trail near American University, which can be a nice detour from the hustle and bustle despite subpar footing. The C&O canal, as another person mentioned, is also an option. I don't care for its monotony and lack of shade, but a lot of people can crank out mile after mile on that thing. Good neighborhoods include Friendship Heights, Tenleytown, Dupont Circle, Cleveland Park, Woodley Park, and Spring Valley. They can be expensive, especially for the younger crowd, but, if you can find anything near Wisconsin and Connecticut avenues in upper NW, I would recommend it. A lot of yuppies live in U Street, Columbia Heights, Eastern Market, and Capitol Hill. Gentrification has made once sketchy neighborhoods into desirable locations. I can't imagine you would feel unsafe running there. And if you're putting in any kind of volume, you will be able to reach running routes like RCP easily.
Southern Montgomery County (Bethesda, Silver Spring, and Chevy Chase) also has some nice spots. The Capital Crescent Trail — about 11 miles — runs from SS to Georgetown and has paved and dirt portions. The former are great for running fast. The Rock Creek Trail (asphalt bike path that's decent for running) picks up where RCP ends and takes you past Rockville.
I know decidedly less about running in Virginia, although I did enjoy a few runs on the W&OD Trail last summer. The Custis and Mount Vernon Trails are also decent allegedly. Roosevelt Island, which is off Key Bridge and across from Georgetown, is nice to run around, although you run out of trail pretty quickly. Arlington is a nice area, and the younger crowd seems to prefer neighborhoods like Courthouse. Other suburbs like Falls Church, Fairfax, and Alexandria are farther out. If you plan on living in VA and working in the District, I would suggest finding something right in Arlington to avoid a hellish commute.
Hope this helps. In my eight years of running, I have rarely, if ever, felt uncomfortable while running (I am male, albeit rail-thin).