We have excellent cheese in America.
We also have some of the _worst_ cheese in the world, granted. If it comes in a spray can or as individually wrapped slices, you should AVOID it. Also, if it doesn't say what _kind_ of cheese it is, you should avoid it. Also, Velveeta is not something that you should eat directly. It's meant to be used together with other cheeses in baking (e.g., in macaroni and cheese), to facilitate creaminess. Good cheese comes from behind a deli counter (which can be in a grocery store) and is priced by the pound. You tell the person behind the counter that you want half a pound of this or that kind of cheese, and they ask you how thick you want it sliced, and you say "thick", "medium", or "thin". (You can also just buy a big chunk, if you're not making sandwiches.)
You can also get shredded cheese in the frozen foods section, which is alright for baking, but the number of varieties available this way is more limited -- typically just mozzarella, a couple of kinds of cheddar, and maybe "pizza cheese" (which is mostly mozzarella with a few shreds of something else mixed in, e.g, smoked provolone).
> Are there any excellent and widely available varieties?
My personal favorite is colby. Other good ones that are readily available everywhere (or at least throughout the Midwest) include mozzarella (particularly good for pasta), provolone (good on turkey sandwiches), several varieties of cheddar (mild, sharp, extra-sharp, white), monterrey jack and its various derivatives (cojack, pepper jack, jalapeno jack, lightning jack), swiss, baby swiss, muenster, brick, longhorn, and grated parmesan. Other good cheeses are available regionally, e.g., here in Galion I can easily get havarti, either with or without embedded dill. If I wanted to drive over to Holmes County (about an hour east of here) I could get all kinds of specialty cheeses. The community where I went to college, about three hours west of here, has a couple of good Latin-American cheeses readily available. When I lived in western Michigan for three years, the grocery stores there always had grated romano.
Note too that the light yellow-orange stuff you'll see that's specifically labeled "American cheese" (assuming you get the real stuff) isn't _bad_ so much as _bland_, which is actually useful in certain situations. Among other things, it's great for feeding to gradeschool children, who often don't like strong flavors yet. Admittedly, it's not what I generally want on MY sandwich, and if Europeans get this stuff imported and think it's the main kind of cheese we have in America, that would explain their low opinion of our cheeses.
If you want to visit a place that doesn't have good cheeses, try Korea. I think cheese may actually be against the law or a violation of popular religious beliefs there, or something.