Belgrade market isn't too big, and the first part, dedicated to poor quality clothes, isn't really worth a look. The better part is further in, where you can find vegetables, homemade herbal teas, and nice seasonal produce.
The historic centre of Belgrade is known in the Serbian language as "Stari Grad" (old town). There are plenty of interesting places to explore: the ancient Roman Cardo, the current pedestrian street Knez Mihailova with its commercial shops, the Kalemegdan Fortress and the Republic Square. Another central plaza is Terazije, near the charming pedestrian street of Skadarlija. You can also visit the nearby fruit and vegetable market.