Ok I finally got it working. My problem was the following:
My home computer has Windows 7 Ultimate, x86. I downloaded and attempted to start every version of SQL Server 2012 for x86 OS, but they all gave the "SQL server media not supported on a X86 system. To continue, run the SQL server setup media that matches your system" error when I attempted to run the setup file.
I fixed it by doing this:
In Add/Remove Programs, click "Microsoft SQL Server XXXX (Latest version you can find I guess, mine was 2012) Setup" and click Repair. It should config something for a few seconds and then nothing... after that, I was magically able to start the 2012 installation, no problem. Must have been some weird files hanging around or something.
Hope this helps someone!