So I did some quick internet research and found out that we had created MyISAM tables when we needed InnoDB tables to link FKs. Then I figured out how to change the tables and how to use the SQL box in PHPmyAdmin to make changes to multiple tables at once. After that I figured out how to use PHPmyAdmin GUI interface to create FK links (it was actually really really easy).
But the image_media table kept giving me a "mysql error 1452" error when I tried to link gameID to the game table, even though the link from characterID to the character table worked just fine. I finally figured out that since we had already been playing with PHP forms, a game with the PK 1 was accidentally deleted, but because we hadn't linked the tables, the record in image_media that said it came from game FK 1 was still there. I changed the record in image_media to the right FK and it worked fine. I even changed the PK in the games table and it automatically updated it in the image_media table.
Yes, I am awesome!