DC offers some of the best transit-oriented urban living in the US. Logan Circle, Dupont Circle, U Street, Capitol Hill, and Adams Morgan combine row-house density, Metro access, and rich retail. The Metro reaches most central neighborhoods and Northern Virginia.
Logan Circle, Dupont Circle, U Street, Capitol Hill, Adams Morgan, Mount Pleasant, Columbia Heights, and Georgetown are all highly walkable. The H Street corridor and NoMa have grown rapidly into walkable destinations as well.
WMATA Metro has 6 lines covering DC and Northern Virginia. Service is generally reliable. Coverage of central DC is excellent; outer parts of the city (east of the Anacostia, much of Upper NW) have less rail access and rely more on buses.
Capitol Hill itself, H Street, Eastern Market, Navy Yard, NoMa, and Union Station all put you within walking distance or a short Metro ride to the Capitol.