Brasov's old town is compact and best seen on foot — you can walk from one end of the historic centre to the other in fifteen minutes. For anything further out, the public bus and trolleybus network (RAT Brasov, ratbv.ro) is extensive and cheap, with about 40 lines running until around 23:00 on weekdays.
Buy tickets from kiosks marked "bilete" near the stops — not on board — and remember to punch them when you get on. A day pass is the easiest option if you're making more than a couple of trips. Inspectors are frequent and will fine you on the spot, so don't try your luck.
The most useful lines for visitors: line 4 runs from the train station (Gara) to Livada Postei in the centre; line 51 also connects the station to Piata Sfatului, a minute's walk from the Black Church. Line 20 goes up to the Poiana Brasov ski resort and has a special fare (around 3.5 lei) — mention it when you buy the ticket, as the normal passes don't work on this route.
For Bran, Rasnov, and other nearby villages, buses leave from Autogara 2. Taxis are cheap and plentiful inside the city; make sure the meter is running.