Note: This document is not finished.
This is written for Ubuntu, LinuxMint, and oter debian derivatives that use apt or apt-get. I have tested it on Ubuntu 16.04 LTS. Also, this installation may not be the minimum. In particular, some of the php files May not be absolutely necessary, especially if you are not running nginx and valet.
I have this guide on Fedora using "dnf" instead of apt should work in most places. A few of the PHP bundles are a bit differentently, but It still worked. The Fedora packages seem to be more inclusive, so all of the important PHP components seemed to be loaded.
Homestead is a useful tool. but it can be overwhelming at first if you are trying to learn both Laravel and Homestead.