Git is a distributed version control system, how it interacts with "servers" is very different from Perforce (a centralized version control system). You don't checkout nor commit to a server, that is all local. A central server (ie. a "remote") can be used to coordinate everyone's copies of the repository using "push" and "pull".
This takes some time to get used to, but it's worth it.
While you can run your own centralized Git remote, some of the advantages of Git will be more difficult to take advantage of, and new users will find it harder to understand what is going on.
I would recommend using something like Gitlab which will provide many of the features of Github including an integrated issue tracker, commit tracking, pull requests, and other features fundamental to a well run project using Git.
Another option, which may be more familiar to you, is to use Perforce's own Git Collaboration tools. (Note: I know nothing about these, I just know they exist).