There is no airport in Brasov (as of 2017), although its construction near the village of Ghimbav was approved back in 2005. Local residents believe that the central authorities do not build it, because in this case the city will take the last tourists from Bucharest. Unlikely that this is true, but it's logical.
All interesting places are located in the very center, but if you need to travel by bus, find here here https://www.ratbv.ro/trasee-si-orare/ map and timetable.
We walked in the old town, it is quite large, beautiful and well-groomed.
There are few hills around the old town, the most interesting is Mount Tampa, where you can climb by the cable car. It works every day, except Monday, from 10 to 18. The mountain is easy to identify by the large inscription of BRASOV.
Getting there: Free parking is available near the cable car (N 45 38.397 E 25 35.601). The nearest airport is in Bucharest. From the airport to Brasov there are buses of the Direct Aeroport company (timetable). Bus timetables from other cities can be found here http://www.cdyservice.ro/, http://www.autogari.ro/.