By bus
Plenty of bus routes service U St, although figuring out where they'll take you can be difficult!
The X3 (http://wmata.com/bus/time...) is a very convenient express route heading west from U St/Cardozo through Adams Morgan to the National Zoo.
#52, #53, and #54 (http://wmata.com/bus/time...) all run along 14th St NW straight from the Mall and up to the Columbia Heights metro stop.
#S1, #S2, and #S3 (http://wmata.com/bus/time...) run up along 16th St NW from the north end of the White House through U St, and on all the way up through Carter Baron and on to Silver Spring, MD.
The D.C. Circulator Woodley Park-Adams Morgan-McPherson Square Metro Station "Green" line runs Su-Th 7AM-midnight, F-Sa 7AM-3:30PM. It has two stops on 14th St, one at U and one at Massachusetts just west of Logan Circle. It comes up from downtown, and goes north through Columbia Heights and Adams Morgan and then on to Woodley Park.
By Metro
The neighborhood is serviced by the U Street/African-American Civil War Memorial/Cardozo stop on the Metro's yellow and green lines, with entrances at the the southeast corner of 13th & U St NW, and on 10th Street, south of U.
By car
By car, the main streets are 14th, R, 9th, and Florida. North Capitol is a good and relatively uncongested artery heading north towards Maryland and the Beltway. Avoid driving on U St, because it is very congested that is, even more congested than the rest of these streets. On-street parking is possible on the quieter side streets, really any time of the week, although if you are not familiar with the area driving in is definitely not recommended. Throughout Shaw, and really throughout most all of north-central D.C., smash-and-grab robberies of parked cars are more common than you'd like.