I got a new computer and am getting rid of my old computer, so I wanted to move my game progress... What a pain. How do you do this?

According to PCGamingWiki (which should be your first stop whenever you're having trouble finding where save or config data is stored :) ), the relevant info is stored in the Windows registry. (If you're playing on a Mac or a Linux distro, that page won't help, as there's no info there about where those versions of the game store the data, and I don't know enough about such OSes to make any guesses.)
I have virtually no experience messing with registry entries, so that's about the most help I can provide. =)