How to introduce my team to work from Github?

I want to introduce working from Github into my team. Currently, we each have a pre-production machine not connected to the live environment. Each of these machines has a directory of Common components, which are generally class libraries that can be dropped into new projects for SQL connections or parsing files. However, if there is any collaboration between developers these classes/projects are copied and passed between each other via memory sticks. I have some experience with Github in that I use it for small personal projects and with this knowledge I have them convinced that it is how we should move forward.

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What I am asking advice on is, how I would organize a small development team so that; Common Components (once one version of each has been agreed upon) can be included into a project without risk of inadvertently pushing changes to the repo of that component. Applications that are deployed to live will not inherit changes from updates to components. And projects that are currently in development do not get lost in common components repos.