"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