Git configuration for local and remote web servers. Where to start?

I am part of a 10-members team and we build websites (dozens of them, ~9gb each). Since the company is old-fashioned we don't use git nor any kind version control, instead we have 3 different kind of servers (usually running windows server):

• local server (in a room near our office) for testing, once we're satisfied with the local testing we push the changes via ftp in a
• preview server (remote), used by the client to validate our new features. If the client is satified, we push the changes via ftp in the
• official server (remote), which is used by the consumer

As it can be seen we are, in fact, using 3-style branches: test, preview and master.

But we do not have a server for each of us:

We access the local server (folders) and directly modify the files, and via the browser we see the changes. Usually we do not access the same file at the same time so we do not have conflict.

I know we should run the server in our personal computer, make the changes we need and then push them in the remote preview and official servers...But we can't do this way, cause we already have 7 different workstation (in the room near our office) running all the local websites.

Maybe we sould: do not change the way we work in local, set up git in all the server so: we push the changes from local server to preview and official server.

So, how can i configurate git to our needs? Are there some kind of strategy i sould study?

thanks in advance