Belgrade has Kalemegdan, but Zemun (right across the river) has Gardoš. Gardoš is smaller but I would say more charming. Kalemegdan is full of tourists while Gardoš is more calm, I highly recommend this place.
It is located in Zemun so it is way different than Belgrade or New Belgrade. It is way easier to feel the charm of Zemun than to describe it :)
Just bring your photo camera and take some amazing photos!