Things to Do in Belgrade

Republic Square.
Republic Square is the central meeting point within Belgrade. It's recognized by the oldest statue in Belgrade, a statue of Prince Michael who liberated Belgrade and the surrounding Serbian cities from the Ottoman Empire in 1867. He then went on to help with the cultural development of the city, having developed the National Museum and National Theater. Prior to World War II, the square was called Theater Square, as Serbia was still a dynasty. It wasn't until after WWII that Tito declared Yugoslavia a Republic that Republic Square got its name.
Skardarlija is the oldest Street in Belgrade, located in the Bohemian Quarter. It still retains the quaint cobblestone and is dotted with plenty of cute kafanas and cafés; it is certainly one of the prettiest things to do in Belgrade. The Bohemian Quarter came to existence thanks to the Gypsies in 1830 and, by the late 19th century, people began journeying over to this area to play and listen to live music all night long. That vibe still rings true today! Be warned though, this is quite the touristy street! In fact, Skadarlija contributes to one-third of the city's foreign currency income!
Kalengaden Park and Belgrade Fortress.
Belgrade Fortress consists of the old citadel and Kalemegdan Park at the confluence of the Sava and Danube Rivers. It has been destroyed and rebuilt for centuries and centuries, due to all of the wars that have taken place over the years. In fact, Kalengaden Park completely covered a military bunker in 1948; it was built because Serbia was afraid that Stalin could attack at any moment. The craziest part about it is that the military bunker wasn't even discovered until 2008, making it one of the most interesting things to do in Belgrade!
Hundreds of years ago, the city of Zemun used to be an entirely different country, occupied by the Austro-Hungarian Empire. When you go from downtown Belgrade to Zemun, it feels as if you've crossed into a different country as it has such a unique feel. In fact, up until World War II, Zemun was just a neighboring city down the river! It wasn't until after the war that Zemun was considered part of Belgrade.
St. Mark's Church.
Despite looking like it was built in the 14th Century, St. Mark's Church was actually constructed in the 20th Century! St. Mark's Church used to be the biggest in the city, and tried to revive the old styles of the Middle Ages. As with most Serbian Orthodox Churches, there are no pews or chairs as the congregation is expected to stand throughout the service, no matter how long.
Church of Saint Sava.
The Church of Saint Sava is the largest church in Serbia, and ranks amongst the largest churches in the world (#3 or #5 depending on your source). It's named after the founder of the Serbian Orthodox Church, and despite beginning construction in 1935, it still is not complete. The exterior is beautiful and dominates Belgrade's cityscape, yet the interior is completely empty. The main dome's mosaic is complete (after the Russians donated 4 million euros to complete the mosaics), but the rest remains underwhelming. Do enter though, and be sure you go down into the crypt, as that's bright and beautifully ornate.