By bus
The following metrobus routes run 5AM-midnight daily, usually with a couple additional buses midnight-2AM:
52, 53, & 54 (http://wmata.com/bus/time...) travel from 4AM north-south along 14th St from the East End through Shaw to the Columbia Heights metro station, and then all the way north to the Takoma station at the edge of the city.
H2, H3, & H4 (http://wmata.com/bus/time...) run east-west from the Cleveland Park Metro station on the Red Line through Mount Pleasant, the Columbia Heights Metro stop, and then past the Washington Hospital Center and on to Catholic University.
The L2 (http://wmata.com/bus/time...) runs north from GWU through Dupont Circle, then straight up 18th St in Adams Morgan before veering west to the Woodley Park metro stop and the National Zoo. Sa-Su service from 6AM.
The D.C. Circulator Woodley Park-Adams Morgan-McPherson Square Metro Station "Green" line runs Su-Th 7AM-midnight, F-Sa 7AM-3:30AM to serve the nightlife traffic. It stops at the 18th & Columbia intersection and the Columbia Heights metro stop, and then goes west to the Woodley Park station, and south along 14th through U St and on to Downtown.
By Metro
The closest stop on the Red Line is Woodley Park/Adams Morgan to the west of Rock Creek Park, which is actually a good ten minute walk from Columbia and 18th street. Columbia Heights on the Green/Yellow Lines is the other obvious stop, and is closer to Mount Pleasant.
By car
It's not very hard to find street parking on side streets in Columbia Heights, and to a lesser extent Mount Pleasant. Adams Morgan, on the other hand, is packed. Since robberies are more common in this part of the city than any other, taking your car here can be a risky venture if you will be walking back to it at night by yourself. Taxis are easy to catch on the main drags, and pretty much anywhere within Adams Morgan. Lots of cab drivers live in Columbia Heights, so they are easier to catch throughout than you would expect for a less lively neighborhood.
The main streets for driving north-south are 16th St and Georgia Ave although 14th can be a surprisingly quick route north of U St. Major east-west routes are fewer and more confusing: Columbia Rd is the best route to go between Connecticut Ave in Dupont Circle and the Washington Hospital Center and Catholic University.