You could do something like this. In this workflow I'm assuming a central server (origin>), your computer (local>) and some devolopers (deva>, devb>).
First lets create some local tracking branches for your dev, staging and live branches and push them up to the central repo. This is only needed once so not much overhead.
local>git checkout -b dev
local>git push -u origin dev
local>git checkout -b staging
local>git push -u origin staging
local>git checkout -b live
local>git push -u origin live
Ok.
Now you have three branches that are setup as local tracking branches to the corresponding remote branches on origin. Now any developer can do the same, but they need less amount of work to do so. First they need to fetch the updated references so we can reference a remote branch and after that create the tracking branches with the git checkout command. This need to be done by all developers that are going to contribute to the central repository (and only once even in this case).
devx>git fetch
devx>git checkout --track origin/dev
devx>git checkout --track origin/staging
devx>git checkout --track origin/live
So lets say that deva has done some work that you want to take a look at. We assume that he/she is working on a new feature on the dev branch.
deva>git checkout dev
deva> ... edit some files
deva>git commit -am "new feature bla bla"
deva>git push
You get a mail from deva that he has pushed his changes. Then you do:
local>git checkout dev
local>git pull
Ok. So now you and the other developers have a system up and running so you can collaborate on any phase of development. But how do we get the changes up and running on the server environment?
We can solve this by utilizing the hook system in git. You can let the central repository notify each server environment that something has been pushed to it (you can easily detect exactly which branch that was pushed) and take action accordingly. This way the servers can update themselves, without you having to care about it. Well, of course you need to be careful what you push and don't push. But that requirement will always be there, no matter what kind of system you have.
Hope this helped you realize your dream :D