MinGW contains The Standard C compiler packaged for Windows. On the download page
http://sourceforge.net/projects/mingw/files/ , I do not see anything about ‘mingw-w64’. What I do see is mingw-get-setup.exe (86.5 kB) which is a program to get the rest of the compiler and whatever other files you want. I’d suggest you get that and see if it provides everything you need. If it doesn’t, it will probably tell you where to find it.
Part of learning C programming like this is learning the procedures and methods. This is not a one-step procedure or compiler.
Also, on the Mac the standard command line compiler is Clang, which is a GCC compatible compiler. You can get this by installing xCode and the command line tools. You can also install GCC, which you choose is up to you as they both (more or less) behave the same (since Clang is designed to be compatible with GCC).
>> How do we feel about mingw-w64, rather than MinGW?
At this stage in his development (pardon the pun) I really wouldn’t get too hung up over the architecture. For the type of code he’ll be writing he really won’t need to know nor care whether it’s a 32 bit or 64 bit compiler. This is super advanced stuff and is highly unlikely to even be covered in his course.