sfg
Certain sites/apps will have different setup tools (e.g - AWS for JumpQ and Dan Harris for MFPA) - And specific instructions should be written in individual repos for them.
But by and large - we manage our own Bytemark servers running on Debian. The deployment tool of choice is Capistrano. It's ace for managing all required tasks that need to be ran on every deployment - as well as keeping 5-6 full backup versions on the server which can be fallen back on should a deployment fail.
A really good guide to setting up the capistrano can be found here, which deals with the full process of hooking up with Git / connecting it to your server / deployment commands.
You can also set up cap configuration for different environments here - so if you need to set up configuration for a separate staging and production site then you can can do.