Are you using source code control? If not, you should. I suggest using Git, for example hosted on Github.
For a simple setup like this you don't need to use any special deployment tools; you can also use Git for deployment.
Developing directly on the staging server is not a great idea. Try to set up a development environment environment on your laptop.
You can push from your development machine to Github. Then then on either staging or the live server you can connect via ssh and pull from Github.
This allows you you to use all the power of Git to create branches and tags and to rollback to an earlier version if you make a mistake.