Most of the city's important sights are more or less within walking distance of each other (which is also the best mode of travel through the city for appreciating the architecture and atmosphere).
The city bus timetables are available here. A single ticket costs RSD90, when bought from the driver. Don't enter the bus on the other than the front door. Lake town Palić is accessible by public transport: bus #6 stops at the 3 bus stop to Palic near Hotel Patria in the city center. You can use the Moovit App to get around by bus.
In case you decide you need a cab (to see some stuff on the other site of town for example, or for whatever reason) there are taxi stations on many locations but the easiest to find for a tourist are probably the one next to the Town Hall and the one next to the train station. Taxis are not overly expensive by Western standards. Make sure the taximeter is on.
If you're in shape you can rent one of the 56 bikes available in 4 parking-points.