First off, forget american cheese. The de-fatted cheese is pretty much like alcohol-free moonshine, or a shaved cactus: pointless. Here, the cheese is full-fat (40% or more of milk fat), semi-fat (around 25%) and processed cheese, aka melted, which most people don't even count as cheese.
Maybe not as diverse as France or Italy, or maybe just not as crafty in self-advertising, Serbia does have many kinds of cheese. It's actually a natural phenomenon, the recipes coming into existence over many different places.
First, the variety of fresh cheeses. Forget cottage and ricotta, there are dozens of different kinds, even the aged fresh cheeses (i.e. they are still white and look like fresh, and they weren't pressed). Also, lots of cheeses from sheep's or even goat's milk.
Next, the hard cheeses. The socialist industry in the fifties developed a set of common recipes, so there are standard kačkavalj (from italian caccio cavale, horse cheese, because it was processed in the mountains and then carried to market on horses), trapist (allegedly from the recipe of trapist monks), ajdamer/edamer/edamac (variants of the same name). They ARE fat - actually, the regulations still require to state the percentage of milk fat in it, and it goes to 45% (so called "full fat" cheese). These are completely unlike the american fat-depleted cheeses, and taste as a cheese should. The main difference is when you cut it with a knife - a proper hard cheese shouldn't stick to the knife, the slice should just fall off.
Then there are the melted spreadable cheeses, and cream cheeses. The former are generally regarded as lower kind, as it was known from the outset (some time in the sixties) that these are reprocessed (i.e. molten) remainders from production of regular cheeses. The cream cheeses are what you expect, with varying proportions of young cheese, butter and cream in the recipe.
Beware, though, when buying. The americanized fashion of making "baby" versions is understood here exactly as it was intended - as a way to sell more water at the price of cheese. The shrink wrapped, semi-soft things marked as "pizza cheese", "baby..." are exactly that, lower grade, less pressed and dried. They may cost the same as the real stuff, so it may take some time to discard some brands. Most of today's shrinkwrapped cheeses are of this americanized sort (except they still contain as much milk fat as the label says*), i.e. they stick to the knife, are too soft, and generally contain more water than a decent cheese should. Just press it lightly with a finger - if it feels soft, then it probably is the wrong kind. Today's kačkavalj is not even a distant cousin to the hard cheese it once was; I've found one brand from Kopaonik to be exactly what it should be, and a couple more that are passable.
Generally, to buy a good one, stay away from the fancy printed wraps. The small dairy shops don't have the money or marketing to come up with anything but a vacuum wrap with a sticker on it - but they make much better cheese.
* But that's IN DRY MATTER, so if you removed the water you would get the 45% of milk fat. It's just that the hard cheese, by its nature, can soak or lose water - so the water doesn't count. It counts when you pay, though - the shrink wrapped cheese will never dry until you pay for it, so you pay for all the water first.