I highly recommend using https://github.com/, they have pretty decent client program and if I'm not mistaken 5 free private repos and unlimited private repos for 7$/month.
They also provide pretty good web dashboard (where you can do things like releases, manage branches, tickets, view readme.md files etc) and overall are super easy for people who are new to git / source control.
Another alternative is bitbucket, but they are much less user-friendly and I wouldn't recommend them for new users, and git lab which I'm not too familiar with but should be pretty similar to github.
Its really important to learn how to use source control and it sounds like a good time for you to get into it. Just be sure you open a private repo if you don't want anyone seeing your code.
PS. MonoGame itself is hosted on github: