On foot
The Woodley Park/Zoo Metro stop is actually the closest stop to Adams Morgan. It's just a half mile walk over the Calvert St Bridge from the station to the heart of Adams Morgan.
By bus
D.C.'s bus system is not always the fastest way to travel, but the network is extensive to the point that you can reach any destination even in this outer neighborhood without a car.
Routes #30-36 (http://wmata.com/bus/time...) (http://wmata.com/bus/time...) all run between Georgetown and the Friendship Heights Metro station along Wisconsin Ave. On the weekends after midnight, the buses run once every hour until 2AM.
#L1, L2, and L4 (http://wmata.com/bus/time...) run the length of Connecticut Ave 5:30AM-midnight daily. The L2 continues into Adams Morgan, while L1 and L4 head further south along Connecticut to Dupont Circle and then on to George Washington University.
Routes #N2-N6 (http://wmata.com/bus/time...) all run along Massachusetts Ave from Dupont Circle to American University.
#D6 (http://wmata.com/bus/time...) is the only reasonable way to reach the Kreeger Museum via public transport, which runs through Georgetown along Reservoir Rd, coming from the Dupont Circle metro station at P and 22nd St NW.
By Metro
The Metro Red Line runs through this part of the city, and it is pretty self explanatory, given that the stations are all named after the neighborhoods and their major attractions. Woodley Park/Zoo for Woodley Park and the National Zoo, Tenleytown/American University for Tenleytown and AU, etc. The National Cathedral, Glover Park, and the Palisades, like their southern neighbor Georgetown, are not served by Metro, and will require either a car/taxi or bus ride.
By car
Driving is another option, though competition for parking spots can be fierce. You can usually find parking on side streets off of Connecticut and Wisconsin Ave, although you might have to walk a good ways from your car, and parking is usually limited to two hours in neighborhoods closer to downtown. The Palisades, along the extreme west of the city, are best visited by car, and it's quite easy to park there.
If driving in this part of the city, be aware that during weekday rush hours Canal Rd in the west becomes one-way and reverses directions during weekday rush hours M-F 6:15AM-10AM inbound, 2:45PM-7:15PM outbound. Similarly, Connecticut's left lanes are reversible, so pay attentionâyou don't want to find yourself charging into oncoming traffic!
Taxis are generally not hard to catch along the main avenues Wisconsin and Connecticut, but will be impossible to find elsewhere.