Trg Republike seems to be at the center of Belgrade and the place where everything and everyone meets. It's just facing our hotel so we passed by it dozens of times during our stay. It starts as a quiet place in the morning, only to become a busy meeting point for everyone as the day goes by.
For a lot of locals this is the center of the city and the meeting point if front of the famous "horse" (monument of Knez Mihajlo on the horse) :)
It is located at the beginning of Knez Mihajlova street and it's quite busy and full of people. Behind the monument of Knez Mihajlo is National museum and just across...
Trg Republike is the real hurt of Belgrade. The National Museum (not open to the public yet), the wonderfull building of the theatre and all the market and cafe streets begin there. You jsut need five minutes to get to Katsarlija, to the Fortress and to the market streets. The statue of Prince Michailo is astonishing and is the gathering...
This are the central streets of Belgrade and they are always full of people. It's very nice to jsut walk by around here, admire the buildings and just walk by. The serbs are generally very sociable and very open to foreigners so that is certainly a plus.
If you need to wait for somebody, it;s the place to go. Consider even to have a coffee over there, but no more. There are definitely more pleasant (and more affordable) places in belgrade than the square.