By Metro
Montgomery County is served by both the northeast and northwest sections of the D.C. Metro Red Line (http://wmata.com). The northwest section includes stops right in the thick of Friendship Heights, Bethesda, Rockville, and Gaithersburg; the northeast section in Takoma Park, Silver Spring, and Wheaton.
By car
Traffic comes in from the northwest on I-270, and pours in from east and west on the habitually clogged Beltway, I-495, which brings in drivers from I-95 in the northeast, and from Virgina to the south. There are no highways leading here from Washington, D.C. and this contributes greatly to the traffic. The major arteries from west to east include Wisconsin Ave, Connecticut Ave, and Georgia Ave.
By bus
Generally it's easiest to take the Metro from D.C. and then transfer to bus from the Metro stop, but there are a few major Metrobus (http://wmata.com) routes to Montgomery County from the city: #30-36 (http://wmata.com/bus/time...) run up Wisconsin Ave to the Friendship Heights Metro station, and #70-71 (http://wmata.com/bus/time...) run up Georgia Ave to the Silver Spring Metro station. From these stations you will have to transfer to another bus to keep going.
By plane
Like the rest of the Maryland Capital Region, Montgomery County is served by the nearby airports of Dulles International, Baltimore-Washington International, and Reagan National.