In the past 20 or so years, Belgrade has become much more accessible. Whether by car, train or plane, many routes have been opened and renewed, making travel much less of a hassle.
AirlinesAfter the local Airline, JAT Airways, was bought by Etihad Airlines, Thus making it Air Serbia, many new routes emerged. Currently, there is 28 airlines commuting to and from 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 18 km 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), your safest bet would probably be 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.