I don't like cheeses that smell (and probably taste like) baby vomit or dirty feet. That rules out blue, chevre, etc.
I agree American is not cheese (don't they call it cheese food product or some such nonsense?). That said, I sometimes order it on a burger - just seems the thing to do sometimes.
Mostly a fan of cow's cheese, I like drunken goat and Manchego is pretty good - especially with dates and figs. Love extra sharp cheddar, gruyere, brie, camembert, Morbier, Compte, asiago - I could go on.
Nothing beats a charcuterie plate with cheeses and crusty bread.