While this is possible, there are a few things to consider:
Disclaimers aside, probably the first thing you should do is download the staff client and connect to one of the demo servers. There is documentation and tutorials. Play around and see if it might work for you. For running your own server, perhaps the simplest step would be to install Virtualbox and run an Evergreen virtual image with it. Otherwise, check out the download and installation instructions.
Evergreen 2.1 and earlier (circa 2011):
See the page for developing code.
The former. There are just a handful of "core" developers that have deep and wide understanding of Evergreen, and they funnel contributions from others in the development community into what becomes the “official” Evergreen. People with different skillsets are found in that dev community; some are expert programmers just learning Evergreen, others are novice programmers helping out with bite-sized problems, and yet others help out in ways not involving “coding” at all, whether with IT expertise, library domain knowledge, testing, documentation, leadership or advocacy (for whatever goal).
Would you be able to recommend a path to start learning more?
A disclaimer: you don't have to be a programmer or have programming staff to use Evergreen, and there are other avenues for getting customization done (such as hiring folks, or an Evergreen company, within the community to do work for you, or contributing in other ways such that folks want to help you in kind–even simply opening bug reports can instigate change). In general, we like to encourage adoption of useful features into stock Evergreen as opposed to encouraging local customizations that never leave their original libraries (and for changes that don't or won't make it into Evergreen proper, it's still useful to sharethem with others).
That said, probably the best thing to do is to start by installing Evergreen (a useful learning experience in and of itself), or at the very least, grabbing a pre-installed virtual image of Evergreen to run inside a virtual computer on your desktop with software like VirtualBox.
Both options can be found here: http://evergreen-ils.org/downloads.php
If you need to remain anonymous (maybe for fear of retaliation from your current library vendor), there are ways to do that and still participate in the community. You can ask at email@example.com for advice in that regard. In general though, it'd be nice if you just said Hi and introduced yourself on the dev mailing list and/or in IRC.
Launchpad is our public bug tracker; you'll want to create an account for it: http://launchpad.net/evergreen
Our mailing lists can be found here: http://evergreen-ils.org/listserv.php
Information on the IRC chat here: http://evergreen-ils.org/irc.php