Zeljko Jovanovic: Get the A1 bus in front of the Arrivals. It goes to the central Slavija square via New Belgrade and the Railway / Bus station. The ticket (just Din 250) is bought on the bus.
Jovan Đukić: Nikola Tesla Airport's best kept secret - Moritz Eis is just nearby the customs and passport control. Head to the Dufry self service cafè to taste the best ice cream there is in Serbia! :)
Ayrat: You can find some free open wi-fi connections at underground floor, just opposite hertz car rental. BTW, there are many rent car offices, international and local ones.