One easy thing you can do is simply zip up your packaged game, go to the “Releases” page on GitHub and attach your file to a new release. That’s how we currently distribute the binary dependencies for UE4.
Use the .gitignore file. Create a file at root of project called .gitignore if its not already there, then add the path to the saved folder you want it to ignore. You can find examples online.
If you need a free private Git hosting that can support larger files, BitBucket is a good choice. It’s limited by number of users, but unlimited bandwidth and private repos are free.
You can actually configure Github for Windows to push to a BitBucket repo.