Cheese for Hamburgers
I like a melty melt on my burger so I avoid those fancy twee concoctions like crumbled goat cheese. I want gooey.
For mild, the classic would be American cheese. For better cheese with equal liquidity, try Havarti, Monterey Jack, Pepper Jack (for spice), Colby, Fontina, young Asiago or young Gouda (don't forget to remove the wax).
Alpine-style cheeses give you glorious ooze but more intense flavor--nutty, roasty intensity via cheeses like Emmentaler, Gruyere, Comte and American offerings such as Spring Brook Tarentaise, Jasper Hill Alpha Tolman and Pleasant Ridge Reserve (these will jack your price tag up).
Blues are also brilliant--the salty, spicy, peppery notes of the cheese contrast with the mellow, sweet beefiness of a good burger. Stick to softer (higher moisture) blues for reliable melting and flavor, such as Cambozola Black, Fourme d'Ambert or Gorgonzola Dolce.
Avoid sheep milk cheeses which will go greasy and hard, dry, aged cheeses that will soften and get oily but won't melt.