I've tried the varieties commonly found everywhere: swiss, mozzarella, American, provolone. I've never bothered to try anything fancier than that.
I don't have any specific preferences.
Mozzarella is a soft fresh cheese. Try ricotta which is great on bread with a little salt. I don't like most supermarket varieties of ricotta, see if there is an Italian or European grocery store that sells authentic ricotta. The better ricotta is worth the price difference and tastes so much better.
Swiss, american and provolone are semi soft to medium hard cheeses. Find a cheese shop in your area that can recommend a good substitute for these cheeses. I like munster, gruyere, gouda, jarlsburg and cheddar.
You might ant to try some hard cheeses. The flavors are more pronounced and sharper. Most cheeses typically known as "grating cheeses" like parmigiano-reggiano and pecorino romano are also quite good on their own when cut into chunks. Their saltiness combines quite well with fruits like apples, nuts, various antipasto items and wine.
Don't be scared of moldy cheeses like blue, gorgonzola, roquefort, brie or camembert.
Brined cheeses like feta can be particularly salty but still tasty. Feta is one of my favorites and pairs well with green olives and bread.
Walk into a cheese shop and see what samples they have to try. Sometimes, cheese are better when paired with certain foods or wines so you may enjoy cheese more if you begin to eat it with an accompaniment.