The Cheese
A grilled cheese doesn't work with just any old cheese. You've got to have a cheese with just the right melting characteristics. Dry, crumbly, fresh cheeses like goat cheese won't melt properly. Ditto for overly aged cheeses like a parmesan or hard Pecorino. For the true classic flavor, nothing is better than ultra-gooey, not-too-sharp American cheese. As we found in our American Cheese taste test, so long as you aren't going for vegan cheese slices, any old American cheese will do. I personally use Kraft Deli Deluxe Singles.
If you want to get fancy, a young cheddar, Swiss-style cheeses like Gruyère (or its French cousin Comté), or young Italian and French cheeses like young Fontina, Tallegio, or Brie go well, too. As long as it melts, it's got a place in our sandwiches.
If you do like the flavor of a non-melter, it's acceptable to treat it like another topping—that is, pair it with a cheese that does melt. A mozzarella and feta combo makes a fine sandwich, as does a Fontina and Parmigiano, for instance.
When at all possible, it's best to go with sliced cheese as opposed to grated. It is easier to distribute evenly, is less prone to making odd holes in the interior of your sandwich, and melts better (many of the best melting cheeses are too soft to grate effectively).