"What to do in Belgrade in your free time" - is not the right question!
The question is "what to do first". The choice of the places or activities is so broad that you will not be deluded:

• Sightseeing - programs with professional tourist guides
   

• Cultural Guide - theaters, cinemas, museums, galleries...
   

• Entertainment - clubs, cafes, casinos...
   

• Shopping - shopping zones, banks...
   

• Tours and excursions - mountains, monasteries, spas...
   

• Sports and recreation - for professionals and amateurs