By Subway
Almost every line that runs in Brooklyn runs through here. The major station here is Atlantic Avenue sometimes called Pacific Street, the biggest station outside Manhattan and a major transfer point. The B weekdays only, D, M rush hours only, N, Q, R, 2, 3, 4, and 5 weekdays only all stop here. There's little to see around it, but if you need to change trains this is the place to do it.
Another major station is Borough Hall, served by the 2, 3, 4, 5 weekdays only, and R. This station will put you near most of the attractions.Clark Street, served by the 2 and 3 is the station closest to the Brooklyn Bridge.
The A, C, F, and G also ply parts of this neighborhood, but transferring to or from them can be a pain although transferring between them can be done more easily. The closest points to transfer between these and the lines stopping at Atlantic Avenue are at Fulton Street in Manhattan which is a very difficult station to navigate, Jay Street/Metrotech a new transfer has been built between the R and the A, C, and F, and the Franklin Avenue Shuttle in Bed-Stuy.
By Bicycle Or On Foot
Both the Brooklyn and Manhattan Bridges can be crossed by bicycle and on foot. Wire mesh limits the view from the Manhattan Bridge, and you will be repeatedly rattled by crossing subways, so the Brooklyn Bridge is much more recommended for pedestrians.
By bus
Many buses connect Downtown Brooklyn with other neighborhoods of Brooklyn see a PDF map here (http://mta.info/nyct/maps/busbkln.pdf, and note the separate inset for Downtown Brooklyn at the upper right corner), but the B51 bus that used to traverse the Manhattan Bridge and connect Downtown Brooklyn with Manhattan was discontinued in the recent round of transit cuts.