Havarti is awesome sandwich cheese.
Port-wine cheddar is amazing. Actually, you can learn a lot about cheese by trying different cheddars. Mild, medium, sharp, extra sharp, a few different types of smoked, a few specialty cheddars, that can be a lot of fun on its' own.
Try some goat and sheep cheese. You may not like everything you try (especially depending on where you get it; some can be very strong) but it's worth a shot. I particularly like goat cheese chevre, a spreadable tangy cheese that doesn't have a very barnyard flavor or aroma. You can often pick this up pretty cheap at Grocery Outlet if you have one nearby.
Manchego is a good hard sheep cheese to try. It's got a little bit of sheep flavor but not as much as some others, so it's not bad as a beginner sheep cheese.
Going somewhere you can taste different bleu cheeses will be a good palate education, as will trying some things like Roquefort and Gorgonzola.
Also check out various treatments of gouda. Different producers make plain gouda differently, plus they'll smoke it or add other elements that make it interesting. I had the chance to try some goat cheese gouda once, plain, ash-wrapped and white-wine soaked. Very interesting, all three.
Try Gruyere, Camembert, Limburger. You may or may not like them but it'll be an education.
