Alternatively, if there's no direct price on using magic (best example I can think of would be Full Metal Alchemist's alchemy, really, because I could say a thing or two about Eragon), then there's just a limitation on the character himself. You began to say it with the Superman-kryptonite analogy, but beyond that, it's like this. Maybe your character does have the ability to read minds or control fire. However, there's two things to that:
1. Every ability has an outside consequence. Just ask Hellboy's Liz Sherman about how being a powerful pyrokinetic who hasn't yet learned to control her powers = burning down half a city block. Most likely, you can't actually master an ability right off the bat, and even then, there's most likely going to be cases of "what would the neighbors think" and "oops, I'm a master, and I still set the orphanage on fire."
2. That ability can't let the character do everything. Abilities are tools. They shouldn't be the only guns the character possesses, nor should they be things that make up an entire character. They help, sure, but you can't just SHOOP-DA-WOOP someone in the face and be done with it, especially if your powers are specific. Telekinesis, for example. You can throw a car at someone with your mind, but I'm pretty sure you can't use it to set a zombie army on fire, regardless of what the Resident Evil movies would like you to believe.
2a. And if your character can't do everything with a handful of special abilities and normal skills, making them a Swiss Army knife by giving them more abilities is doing it wrong. But that's a tangent, actually.