This is the city center of Belgrade. This is the most ideal place to stay and if you are doing some shopping, you'll find most of the stores here.
Serbia is all about the fortresses, you could check out both fortresses in Belgrade and Novi Sad. In Belgrade, you have the biggest fortress in the country - Kalemegdan.
If you want to try the most authentic Serbian food along with some live Serbian music, Skadarlija is the best place to go. Most all of the restaurants here are good (just look for the busy ones) and they're all a great night out. Make sure to order some Rakija!
Novi Sad is a small city but the church will always help you navigate around. By the main square you have the Novi Sad theatre which is a really nice street. Then next to the church to the left is a little square with nice bars. Up to the right is all shops and restaurants.
This fortress isn't as big as the one in Belgrade but it is a great sight to see in Novi Sad. On the top there are three restaurants all of which are also good to have a coffee or a couple of beers. If you're going up here, make sure to go out to the viewing point to see out over the Danube!
One of the nicest parts of Novi Sad is around Dunavski Park. There are some nice bars and restaurants right by it and in the middle of the park is a great café.
This is a man-made beach in Novi Sad that has some nice bars and restaurants next to the water. If you wanted to put in a day of beach volleyball or basketball, this is where you'd go.