How to back up a Chromebook? Do nothing. That is how simple it is. The backup occurs automatically in the background and stored in your account in the cloud.
Even setting up a Chromebook is not that difficult. See the video below:
SyncToy, a free PowerToy for Microsoft Windows, is an easy to use, highly customizable program that helps users to do the heavy lifting involved with the copying, moving, and synchronization of different directories. Most common operations can be performed with just a few clicks of the mouse, and additional customization is available without additional complexity. SyncToy can manage multiple sets of folders at the same time; it can combine files from two folders in one case, and mimic renames and deletes in another case. Unlike other applications, SyncToy actually keeps track of renames to files and will make sure those changes get carried over to the synchronized folder.