Following this tutorial by Justin you can setup a production server that uses git hooks for deployment based on commits to the master branch.

Repeating that process on another server (or the same one if you’re short on droplets) and changing the post-receive hook to watch for commits to the staging branch instead of master .

Lastly I would add a remote such as:

 git remote add staging user@staging.website.com:repo

Automating the push from staging to master will depend greatly on your workflow, automated testing, code review and similar things specific to your repository / project.

For small projects and applications I add my automated tests to the post-receive hook, if all tests pass it will tell me in the response of git push staging <branch> so I can then merge to the master branch and push to production manually.
Additionally, commits against the hotfix branch on the staging server are automatically merged to master and pushed to production if all of the tests pass, if for any reason tests don’t pass on the production server it will revert back.

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