By Subway
The subway is overall the best way to get around Brooklyn, although there are some spots, notably Red Hook and parts of East Brooklyn including most of the tourist attractions, where subway coverage is poor and bus transportation is encouraged.
The biggest hub is Atlantic Avenue also called Pacific St. From there, there are transfers to pretty much any line in Brooklyn. Many of the lines that don't stop there do stop elsewhere in Downtown Brooklyn. After Downtown, however, the lines fan out in all directions, and it is difficult to transfer elsewhere, although the Franklin Ave. Shuttle in Bed-Stuy does help somewhat. The G can transfer to the lines that don't stop in Downtown. In East Brooklyn, the L serves the same purpose.
Broadway Junction in Bed-Stuy and East New York is a minor hub between trains that don't stop at Atlantic. Changing there can be useful if you're going to Northern Brooklyn. Coney Island is a major hub for four lines--it's possible to do a "loop tour" of Southern Brooklyn by changing there.
By Commuter Rail
There is the option of taking the Long Island Rail Road if your starting and ending points are in the following neighborhoods: Downtown Brooklyn, near Atlantic Avenue/Flatbush Avenue, Bedford-Stuyvesant, near Atlantic Avenue/Nostrand Avenue, and East New York, near Atlantic Avenue/Van Sinderen Avenue. It isn't terribly convenient, due to the limited stops and relatively infrequent service, but if you need to go to exactly the areas it stops in, it's faster than the subway.
By bus
Buses generally fan out from one of several hubs, including the Coney Island-Stillwell Avenue subway station, Kings Plaza, the Broadway Junction subway station, Downtown Brooklyn, the Myrtle/Wyckoff Avenues subway station, the Brooklyn College subway station Flatbush Avenue, and the Williamsburg bus plaza.
There are locations, including Kings Plaza, where dollar vans shuttle riders to and from nearby subway stations.