I've tried downloading the MinGW compiler from here: http://www.mingw.org/category/wiki/download . In that folder there was a file called "x86-mingw32-build.sh". I tried setting the path Build & Run -> Compiler to the path that that file is located in, but that didn't work. I also tried setting it to "/home/user/Qt/5.2.1/gcc_64/bin/qmake", which didn't work either. I'm at a total loss. If anyone who knows something about this could help me I'd really appreciate it.
Right, so the issue is that you are trying to work with a cross-compiler for Windows, whereas - based on your comment -, it seems you simply want to build software on Linux for Linux, i.e. no cross compiling.
In that case, you will need a compiler like gcc or llvm/clang. Both should be available from your repository and could install them as follow:
sudo apt-get install gcc
or
sudo apt-get install clang