You want crime? We can arrange some for you.

What are you riding? Road, hybrid, tour? And what kind of mileage did you have in mind?
Probably want to stay with the main trails.
Days mostly. Some of the trails close at sunset. You could do an evening around the monuments.

If you're here on a weekend, a combination of the Capital Crescent and Rock Creek Park. 21 mile loop. I like to start in Georgetown, on K Street. CCT starts under the Key Bridge. The loop goes by the National Zoo, if you want to stop in. On weekends Beach Drive is closed to traffic, becomes a 25 foot wide bike trail.

For more nature, you can go out the C&O Canal to Great Falls. 28 miles r/t from Georgetown. Gravel trail but well worth the trip.

Potomac River Trail from Georgetown, past the Kennedy Center and Lincoln Memorial and out to Haines Point. Some of the best views, especially if you combine it with a ride down the Mall, but the Mall can be crowded.

Mount Vernon Trail goes down the Virginia side of the river from the Key Bridge to Mount Vernon. 19 miles length. Good river views and goes through Old Town Alexandria. Popular and crowded so best to go on a weekday.

Custis Trail is a connector route along I-66, but it gets you to the W&OD at milepost 4. Whole trail in 45 miles long. Mostly flat. Part suburbs, part rural. Best part is past mile 35, past Leesburg. Also good past Vienna miles 12 to 16

Or if you just want to see the city, just pedal around downtown on a weekend. Go around Capital Hill, up Pennsylvania Ave, maybe through Chinatown or up 14th Street and over to Dupont Circle.

Good place to start http://bikewashington.org/