Workflow on Development and Production Servers for a Consistently Updating Website

Question

We have three developers, one system administrator, and an artist that primarily work on a single website (forum) on our spare time to consistently develop features for the forum (but there are other projects that we work on). Because our system administrator recently joined, we dropped managed hosting for a single server and decided to rent out two unmanaged servers (one for testing and the other for production).

In the old server, we simply used Git as a middleman for pushing updates from the non-unified developer team into the server without conflicting updates. Pull other developer updates. Push our own updates. Revert if something breaks.

Since we have two servers now, we plan to push updates to the development server and somehow have it push updates to the production server.

Developer(s) -> development server (bare) -> production server

We want to keep the repositories in the development server and the work tree of the web site on the production server (web server). Is there an efficient way to do this without pushing from the dev server to production? Is there a better workflow for two servers in general?

P.S. The development team consists of high school teenagers and some college kids that have never developed on teams for for-profit businesses.