Thanks for the replies. After some more research, I managed to get the build to work with the glew packages from Fedora 21.
For any other Fedora users that have the same problem, here's what I did:
First, I downloaded the libGLEW, glew-devel, and glew packages from
here. I then tried to install them with "yum install <filename>", which failed because I had packages installed which depended on glew 1.9.
As a workaround for installing the packages in /usr/, I extracted their contents into my home folder using the default system archive manager. Next, I changed the line "include_directories(${GLEW_INCLUDE_DIR})" in "stk-code/CMakeLists.txt" to point to the "include" directory under where I had extracted the packages (e.g., "include_directories(/home/username/bin/glew/usr/include)"). After doing this, I was able to compile the game by running cmake and make with "LD_LIBRARY_PATH=/home/username/bin/glew/usr/lib64" and "LIBRARY_PATH=/home/username/bin/glew/usr/lib64". I then launched the game with "LD_LIBRARY_PATH=/home/username/bin/glew/usr/lib64 ./supertuxkart"
That's all. If you don't have anything that depends on glew 1.9 though, you should be able to completely skip the second half. Also, it might still work if the only environment variable used during compilation is "LIBRARY_PATH" with make.
Now, I'm by no means an expert in doing this kind of stuff so feel free to to post any corrections or improvements if there's something I've missed or done inefficiently.