I don't often write up trip reports, but this forum does not have many, so I thought I'd add my experiences and impressions from a recent trip. I had intended to stay in Belgrade for longer and even visit elsewhere in Serbia, but my plans changed and I ended up staying only 2 days. As for any big city, I could have certainly stayed longer and explored more, but 2 days did give me a good taste of the city.
Overall, I enjoyed my time in Belgrade and was impressed with a city that was lively, dynamic, yet very easy-going. Everyone was very friendly, and many people I encountered were willing and able to switch into very good English. (I tried to master some basics in Serbian but it seemed a lost cause trying to use them.) The cafe culture was very appealing, and the prices everywhere very low, at least compared to the USD. I arrived during a heat wave, which slowed my sightseeing, but I enjoyed just walking around soaking in the city, the culture, and the architecture.
To be up-front, for a capital city its size it does not have a lot of conventional tourist attractions and will not be for every traveler, but I was entranced by the combination of older Austro-Hungarian-style and Orthodox architecture, postwar modern buildings, and contemporary construction. Moreover, I felt like I was practically the only tourist in the city, especially since during my first day I encountered no others. If you like cities and history, there will be something in Belgrade to fascinate you.
The first day I explored the area around Kneza Mihaila and then went over to Zemun for the afternoon - highly recommended! That evening, I walked through Old Town and to Kalemegdan. It's remarkable to see families strolling about late at night and old men playing chess in the park. The next day, I explored Vračar, a seemingly more middle-class neighborhood which I enjoyed a lot. I made my way down to St. Sava (impressive) and Kalenić (not as impressive) before spending the rest of the day in Old Town.
Given my short stay, I did not focus on museums, though I did duck into the Ethnographic Museum. If you're a person who finds the idea of cases of folkloric costume dull, this museum probably won't change your mind! It was interesting to me, though, as an insight into the city's historical capital-city role in Yugoslav national culture, and it's an amazing building.
I stayed at the Vila Terazije, which I would recommend for its price and excellent location. Do note that it is a guesthouse rather than a hotel proper: accommodations prices in Belgrade, while reasonable, do not seem to be especially cheap.
I used the buses some. I took the A1 bus from the airport, which had the advantage of being easy to find at the terminal. It was an older, un-air-conditioned minibus, though, going through a particularly unappealing stretch of industrial suburbia. If you take it, be sure to get a good map to find your way to your hotel. Signage in general is pretty good in Belgrade, but I found the big intersections a little overwhelming and chaotic.
Returning to the airport, I took the 72 bus from Zeleni Venac, which had a more interesting route through residential suburbs. Buses work by refillable paper cards. I got mine at the MyKiosk stands - some didn't carry them, but I would just ask "BusPlus?" and they would point me to one that did. They would ask me how many tickets I wanted and would fill the card with that amount.
For food, I relied mostly on street food, burek and sandwiches, but for a nicer meal I dined at Šaran, a seafood restaurant in Zemun. I wouldn't consider it the absolute best seafood I've ever had but it was all quite good, and reasonably priced. Jet-lagged anyway, I was able to go around 5PM and thus enjoy dining on the terrace without a reservation. A very pleasant meal.
Smoking is ubiquitous, and I don't think I would have enjoyed visiting during cold weather, when one would need to eat indoors.
Thanks to forum contributors here, who gave lots of information I found helpful. I also consulted the Bradt guide and Lonely Planet Western Balkans guide. The Bradt was by far the more helpful, with good maps and a greater insight into the city and culture.