> I just don't even know where to begin. Is this something I could stand up locally and push to like s3, spin up a new server and install 1 thing and have it pull the configs and software installs down?

Yes. You could use shell scripts, Ansible, Salt, Puppet, whatever. I'm fond of Ansible or shell scripts, depending on complexity.

Start here: http://www.opsschool.org/en/latest/config_management.html