I am looking at github but it's making me worry that whoever tries to download my code will have to have git installed and know how to use it in order to download and play my game.
You can also upload 'releases' of a project, e.g. if you want to create an installer that bundles pygame with your application. Again, people don't need to know git to use it.
I think the other major code hosting sites like Bitbucket and Gitlab have similar features. Github is the most popular at present, though.