Belgrade (Beograd in Serbian) is the capital of Serbia, home to around 2 million people. The main motorway (Autoput in Serbian) runs pretty much close to the centre of the city from the North West to the South East. Most cars also use this as a normal road so hitching out of Belgrade is quite difficult.
OPTION 1: First, from Trg Studentski (Students Square) or Trg Slavia (Slavia Square) get bus number 31 all the way eastbound (ie, not to Trg Studentski) to the end of the route (you should see a lot of market people selling fruit on the right side of the bus). The bus closely follows the route of the motorway. From here, you leave the bus and it's a 10 minute walk straight down hill to the actual motorway.
From here you have 2 choices, you can either go west along the motorway to try to find an entrance ramp, or alternately 2km east is a petrol station, however, you have to either walk along the motorway, or through the grass on the verge, however, this gets very thick at times - it's pretty easy though, just hard work :) And take care, on the way to the petrol station is a police station at the highway. 500m before the petrol station you also find a ramp, but with extremely low traffic. The petrol station is build on a hillside, so don't expect trucks to stop there, Serbian trucks seem to have problems enough to go uphill.
OPTION 2: This option involves less walking amongst the highway. From the city centre, take tram 7 in the direction south-east until the last stop (takes about 15 minutes, depending on the traffic). From there take bus 308, direction Veliki Mokri Lug (sometimes written as V.M.L.). After about ten minutes you will cross a bridge going over the highway; get out at the next stop 100 meters later. From the bridge you will already see the gasstation where you want to go, which is about 200 meters walking. This gasstation is also located uphill.
A sign saying "10 K.M." might be helpful in catching a local car which can take you to the first toll stop (a hitchhiking stop regularly used by locals) and out of the intense exhaust fumes.
If you're heading for Nis (or beyond) try using a sign (saying Nis). If going beyond Nis remember to leave the car *on the motorway* - not in Nis, unless you're a big fan of walking. About two hundred meters before the highway split Nis and Sofia (Bulgaria) is an abandoned toll area where traffic is going slow and there's place for the cars to stop.