Yes, unfortunately XNA won't install unless you have VS 2010. Once you have it, it will actually work just fine with 2012/2013, you just need to copy the installed files over and update the .vsix manifest to allow the newer version of VS.
See XNA for Visual Studio 2013 or http://dementedvice.wordpress.com/2013/10/21/let-me-explain-install-xna-on-visual-studio-2013-and-2012/ for more instructions.