This is going to sound extremely food-snobby of me, but preshredded "Mexican" cheese has no place in Mexican food. There are a wide variety of real Mexican cheeses that are useful in all kinds of ways, and they're worth trying out. Here are a few of my favorites; you can find these at many supermarkets and nearly all hispanic/Mexican grocery stores.
Queso fresco: This is a crumbly, unaged cheese, very similar in texture to feta, but with a much milder flavor. It's great crumbled over tacos or scrambled eggs, but it doesn't melt well, so it's a bad choice for quesadillas or enchiladas. Queso fresco's creaminess counters spicy flavors well; one of my favorite snacks lately is a toasted tortilla with queso fresco and hot pepper jelly (call it Southern meets South of the Border). Larger Mexican markets often make their own. If you're lucky enough to live near one of them, fresh queso fresco is fantastic. Aged queso fresco is called queso añejo and has the same crumbly texture but stronger flavor.
Queso oaxaca: Oaxaca cheese is almost identical to mozzarella. It's made by stretching and kneading the curds, and is usually shaped into balls or knots. Grate or shred and use in quesadillas, on enchiladas, or over beans. Oaxaca cheese can also substitute for mozzarella in lasagna or anywhere else--the flavor is extremely close, with a bit more tanginess.
Queso chihuahua: A firmer cheese than either of the above, chihuahua is made to melt smoothly and evenly. Pale yellow in color, its flavor can range from very mild, in between mozzarella and jack, to stronger, like a medium cheddar, depending on how long it's been aged. Use chihuahua in chiles rellenos or queso fundido, a zesty cheese dip. It also makes for super-flavorful quesadillas if you're a cheese lover.
Queso para freir: Literally meaning "cheese for frying," this is a dry, harder cheese that doesn't melt well. It's cut into slabs, breaded, and fried, similar to Greek saganaki, for a great snack or hors d'oeuvre.