It's been years since I was in Belgrade but the Kalemegden Park/Fortress is a must, as is St Sava Cathedral, though be mindful of when you visit and exit walking backwards out of the church. The National Gallery was closed when I was there but I follow it on FB and it looks excellent. The ethno graphic museum was pretty good also. As for food I like Balkan food a lot, the best place I visited was simple called '?' Don't know if it's still there. If you can visit some of the more modernist architecture it's well worth the time. You can always head to Savamala for the hipster vibe. When I first went to belgrade back in 2006 the three carrots pub was the first place I went to orintate myself. Friendly place. Of course if you want night life head to the barges on the Danube, real fun but open to early hours.
I haven't been back since 2012 and then it was full of Americans like Prague in the 90's. Nowt wrong with that, but it's tiring listening to some young un's talking about their novel or finding themselves. I imagine the prices are higher than I was there last. Oh I found taxi drivers to be really honest, except then one coming from Tesla Airport.
Enjoy it, it's not beautiful but it is something else. I love it.