Backing Up Before Upgrade

So this problem that I have has not been resolved by any of the possible solutions that I have searched for (my system freezes on when the graphical environment launches, presumably a package problem with my graphics driver but a purge and reinstall did not work among many other solutions). I have given up at this point and have decided to just do a clean install preferably of 16 but I am on 14.04.

My main conce is not maintaining the installed packages through the update (as this does not seem possible with the advent of snap) as it is to preserve my file data (i.e pictures, scripts, documents, etc.). I have a flash drive large enough to hold said files but no means to easily transport between filesystems.

What I need guidance on is how should I go about backing up these files to the flash drive (through command line or live disk) and then doing a clean install of 16 over 14? (I also have an empty partition that I would like to expand the install onto. The partition is on the same drive)

P.S. I have Windows 7, Ubuntu 14.04, and Kali Rolling all on this drive (it's a laptop)