I know nothing about the GitHub client, but from the git command line you just run the command that is displayed on the Sigil source website, of:
Code:
git clone https://code.google.com/p/sigil/
Change the URL of course if you have made a remote clone of the Sigil respository and want to point to that instead.
If you are on Windows I use TortoiseGIT, mostly because I also use TortoiseSVN so I'm used to that style of explorer integration. It can do more than everything you need, though it can get horribly confusing to begin with the myriad of options. I think John summarised it well in an email when he said git is not so much a learning curve but rather more like leaping off a cliff.