I've been there three times, albeit 20 years ago: for five weeks in Belgrade in the summer of 1983 (weekends in Dubrovnik, Split and Sarajevo), for 4 days in Belgrade in spring 1985 and for two weeks in Dubrovnik in the fall 1986.
My advice is to forget about sightseeing in Belgrade. There's an old fort at the fork in the river, and a walking street with some atmosphereic outdoor restaurants but otherwise absolutely nothing worth seeing. If there is still a Tea Dance Orchestra at the Moskva Hotel Cafe, it's worth seeing for a trip in a time warp and a smile.
Instead, hop a flight (they are probably still very cheap) to Dubrovnik. This is perhaps the best-kept tourism secret in Europe: an almost perfectly preserved medievil fully-walled city on the sea that continues to be a thriving town with something to see at every turn. When I win the lottery, the first three places I will visit will be London, Venice and Dubrovnik.
If Dubrovnik doesn't work for any reason, Split or Sarajevo, in that order, would be next bests. After them: Trogir, anywhere on the Gulf of Kotor, or Sveti Stefan. All are an hour or so flight from Belgrade (plus a short-ish bus ride in the case of the last three). All but Sarajevo are on the coast, can be easily seen in a weekend (the last two are more seaside resorts than sightseeing destinations), and are well worth whatever hassle and expense required. Frankly, almost any place on the Dalmatian Coast is worth visiting, it has to be the most beautiful coast on earth.
And no, I'm not from there, nor or any of my ancestors!
If you take my advice, please post links to photos, I'd love to see how these places look today.