Central Białystok is small enough to cover on foot — almost every sight is within twenty minutes' walk of Kościuszko Square. The Branicki Palace is at the south end of the centre; the cathedral and Orthodox cathedral are on the western side; the Zamenhof Centre is a short walk to the northeast.
City buses cover the wider city and the residential outskirts; you buy tickets from machines at major stops or in the buses themselves. Fares are around 3.50 zł for a single ride.
Taxis are cheap and Uber/Bolt both work in Białystok. For day trips into the Podlasie region, you really need a car — public transport to the Tatar villages and the Białowieża forest exists but is slow and limited.