You're in a pretty safe area along Connecticut Avenue. West of Conn Ave, Kalorama, is a good area. As you go farther east of Conn Ave you can get into some iffy parts of Adams Morgan. Stay alert, but that's always a good rule.

Columbia Road is the main street of Adams Morgan. Busy night life area.
Dupont Circle is also a good busy area. I've been there at night many times, never had a problem, never felt threatened.

Woodley Park, around the Zoo, is also a good area. Lots of restaurants. The neighborhoods west of Rock Creek are mostly good ones.

You'll be fairly close to one of two Metro stations. Bikes allowed at certain times.

To get to Rock Creek Park for riding, you can go thru the zoo, or take the service road on the north edge of the zoo (big hill). If you are here on a weekend, Rock Creek is a very nice ride. Several of the park roads are closed to cars. I highly reccomend the Rock Creek / Capital Crescent loop.

For biking downtown, use either RC or take Connecticut Ave south and use 22nd or 21st streets to get around Dupont Circle. Takes you into the West End, nice area. Several good cross streets to choose from: P, L, I, Pennsylvania Ave.

You've never said what kind of riding you want to do!