I don't know who you've been talking to, but they don't seem to know much about DC (or their knowledge is stuck in the 1970's). Not Lake Wobegon, but nowhere near as bad as the south Bronx (apologies to the Bronx, but I've ridden there a few times)
DC is a great place for getting around by bike, by foot and by mass transit. Very visitor friendly.
I’ve been riding around the DC area for over 30 years. There are a few areas that I won’t ride in the day time although there are several that I’d avoid at night, on bike or on foot. I stay out of Trinidad, although I ride the edge of it often. I don’t ride through Benning or Deanwood or Capitol Heights, but that’s more because I don’t find myself out that way (and you probably won’t either).
This is a congested urban area, so you do have to be very aware of your surroundings. When you go into an unfamiliar area, use common sense. Don’t flash money around. Don’t look too much like a tourist. Remember that you are visiting someone else’s home, so be respectful. If you end up in a neighborhood or situation that makes you uncomfortable, get out of there and back to the main road where there are more people. Traffic can be heavy, so use good road skills.
Northwest and southwest are fine. Capitol Hill is mostly fine, although it gets a little rougher as you go north to Trinidad and east to the river. Northeast is mostly fine above the Arboretum. In Anacostia I ride the Good Hope, MLK, Alabama Ave areas often and have never had a problem in the daytime. I haven’t ridden much in Greenbelt, but I do ride College Park, Berwyn Hieghts, Riverdale, Hyattsville, Mount Rainer, etc, often and with no trouble or concern.
For the areas you’ll probably be visiting, you should have no problem. Georgetown, Northwest, downtown, the Mall, Capitol Hill - all pretty safe and bike friendly areas, depending on road conditions. The city is crisscrossed by bike lanes and bike routes, and many of the non-signed roads are also fine. The Mall and downtown are especially easy to get around by bike.
Greenbelt Park sounds like a good location to be based out of.
Greenbelt Road gets you to the trail head for a nice bikeway into the city using the Anacostia trails. NE Branch Trail to the NW Branch trail gets you close to the city. Rhode Island Avenue or Bladensburg Road will take you downtown quickly, but both can have traffic to deal with, especially on weekdays. I like to take the NW Branch up to 38th Street, then Allison, Varnum and 18th street to either the Arboretum/West Virginia Ave, or Monroe to Catholic U and 4th to downtown. Here’s the link to the Anacostia system
http://www.pgparks.com/Your_Parks/Tr...ail_System.htm
Easy ride to the Greenbelt Metro (Crescent to Kenilworth to Cherrywood). Bikes allowed on Metro except at rush-hours. You could also ride to the station, take the train in and use Capitol Bike Share to get around.
Just north of you is great riding in the Agricultural Research Center grounds and you can go out to Patuxent, Bowie and points east for some nice countryside. College Park might be nice to check out, too.
Get a DC bike map. I think you can order it from the city, but here’s the pdf link
http://dc.gov/DC/DDOT/On+Your+Street...pg,+8.33+MB%29
Here is a good link for local trails -
http://bikewashington.org/
Some recommendations -
Mount Vernon Trail to Alexandria or Mount Vernon
Capital Crescent Trail / Rock Creek Trail on weekends (when Beach Drive is closed to cars)
C&O to Great Falls (I usually take MacArthur Blvd back)
Hains Point especially at sunset.
Seeing the monuments by night is a special experience.
Have fun