In my opinion, there are 3 must-sees in Belgrade. You simply cannot miss them if you're in Belgrade, even for only one day. Kalemegdan, St. Sava Temple and Skadarlija are the "holy trinity" of the must-see in Belgrade .
Good thing is that visiting those places would give you a chance to glance at other attractions, like Knez Mihailova (pedestrian zone), on your way to Kalemegdan, or so-called Silicone Valley (Strahinjića bana Street, with numerous cafes, very popular among younger population), where you can have a drink or two, before you go to have a dinner in Bohemic Quarter of Skadarlija. On your way to St. Sava Temple, you might pass by destroyed buildings during NATO bombing - not that I think it's an important tourist attraction, but I've noticed that all tourists are very curious to see them.
Kalemegdan is a fortress, with quite large park, which accommodates one fortress, 2 churches, one gallery, 2 observatories, 2 museums, tennis and basketball terrains, archaeological remains from several different historic periods, small amusement park for children, two restaurants and one zoo (I'm sure I've missed something, but I guess this gives you the idea). Therefore, you should plan at least 2-3 hours for this one, if you want to see everything.
Although completely out of the way, but not too far, is the St. Sava Temple. It is not finished yet, the interior is still far from finished, but it's open to the public, and there are interesting things to see even inside the church. What you should know is that St. Sava Temple (or if you prefer to call it Cathedral of Saint Sava) is THE biggest Orthodox church in the world, by its area and volume (it's not the tallest). It is surrounded by nice park with fountains, next to the National Library.
Back to the very center of the city, you should plan your dinner in the bohemic quarter - Skadarlija Street. Skadarlija is also known for its excellent restaurants, offering national food, and live acoustic music called "starogradska muzika" (can be translated as "old city music") and the bohemic spirit, remained from numerous writers and artists, back in the 19th century, who made this street famous.