Minibus A1 - goes at roughly 25 minutes, and takes about 30 minutes. Has only 2 stops - one in Novi Beograd (Pariske Komune), and one in Trg Slavija. The ticket is bought on the bus directly and costs 400 RSD (~3€).
Bus 72 - goes every 35 minutes, and goes until Zeleni Venac (Belgrade city center). Tickets can be bought directly with the bus driver, or you can pay at the validator inside the bus with the MasterCard.
Taxi - pick up a taxi voucher inside the airport. It should be around 2000-3000 RSD (17-25€).
In general, taxis are relatively cheap so you can use them for commuting. We highly recommend using Naxis (the number for calling +38119804), Pink (+38119803), Lux Taxi (+38119944), or CarGo (the Serbian version of Uber). There is no Uber or Bolt in Serbia. Beware of the "wild taxis" - they will charge you much more - that's why it's best to use the ones recommended above.
Public transport operates relatively well, although often slow due to the high traffic in Belgrade. The tickets can be bought at the drivers' (150 dinars) or can be paid with MasterCard on the validators in the buses (90 dinars). Details can be found here.
We use Serbian Dinars (RSD) in Serbia, 1€ is about 117 RSD. The exchange offices are called MENJACNICA.
As Serbia is not in the EU, beware that you will most likely be in roaming. Therefore, turn off the mobile data and use only WiFi, or buy a pre-paid SIM card if you prefer.