It depends on what you want to be programming. I develop homebrews that actually run on the Wii U hardware, but others do remote game modding and write up communication tools. The latter you can really do in any language you like (most pick Python) but if you want your code running on the Wii U you have three choices - C, C++ and PowerPC Assembly.
Nobody likes Assembly (for... some reason) so you'll probably end up doing most of your work in C or C++. You can learn whichever you'd like - C++ is more focused around classes and is a more high-level language (sort of like Java) while C is much closer to the metal. In practice, you'll need to be comfortable with both but they're basically the same anyway. If you're taking a course I'd recommend C++ since it'll teach you the basics of C, however you will need to know about pointers and memory on a far lower-level scale than I'd imagine a C++ course would teach (no idea though, never really done one.) Of course, you can teach this to yourself later if you wish.
As for which language is used in reality, I find a lot of stuff is in C++ because of a set of libraries written up by Dimok that make using some of the Wii U's more powerful features within reach for the rest of us. These libraries are written in C++ and it's generally easier to write your program to match. That's only if you want to use those libraries, however; C is perfectly sufficient if you're willing to get your hands (and computer) dirty in the Nintendo libraries.
Personally I've always liked C and try to write as much of it as I can. This is mainly because it's closer to what the Wii U is actually doing than C++. With a bit of work, I could probably translate one of my C programs into machine code: Each of its concepts are reasonable to map to what's going on in the hardware. On the other hand, I'd have no hope of doing this for C++. On a system that likes to crash as much as the Wii U, that's a big thing for me.
To sum up: C and C++ are pretty much interchangeable. C++ is more feature-packed and has good support from the community, but C is easier to debug and (to me, at least) is a lot more logical in the way it translates onto the hardware, even if that means writing code that's a bit more crazy. It's up to you to figure out which is best for you: both would be useful to the community.
Also worth noting that C++ is more likely to get you a paying job if the Wii U doesn't work out.
As I mentioned before, if you're not planning on making homebrew that runs on the hardware, you have free rein over what language you pick.