After the UAE-based Etihad Airlines bought a stake in the venerable JAT Airways, the latter transformed almost overnight in a modern operation now known as Air Serbia, with many new direct routes opening in recent years. Currently, some 28 airlines service Belgrade, a lot of which are low cost. Most European cities are covered by these, yet there are still many more planned.
The local airport, Nikola Tesla, is 18km away from the city centre, making it very accessible. Although there are a few bus routes to the airport (the A1 is the most frequent, with stops in New Belgrade, Savski Trg and Slavija Trg, and the 300din fare can be paid to the driver), the easiest option is taking a taxi. Just make sure you secure one through the official taxi stand in the arrivals gate where they will give you a fixed fare, ensuring you don't get ripped off.
The less known and much cheaper airport in Serbia is Constantine the Great in Niš. And although you can enjoy the cheap transportation via Ryanair and the likes, if your final destination is Belgrade, you will have to add another 3 hours to your journey.