Please do not follow old guides.
as I have heard that it is faster and a little more stable than Ubuntu
Says who? I doubt it. Ok, Mint might use different default software but if that is the case and there is a bug in the software we use you can install those versions in Ubuntu too. Libreoffice, Firefox will be the same versions, with the same amount of stability.
You might like the official Ubuntu MATE. That is the official version based on the old Gnome2.
As I understand it, Mint is basically a distribution of Ubuntu,
No. It is an unofficial copy of Ubuntu where someone changed some of the code to make it their own. We do not support unofficial releases by the way.
They basically tell me to use Synaptic Package Manager
That piece of software has been obsolete for about 3 years or so. That link (did not click it ;) is probably from before 2012. Please do not use old guides. Focus on guides that are at least about the latest LTS (ie. in this case 14.04). Those tend to be correct for at least up to the next LTS.
One of the guides wanted me to copy everything in /Home, which is like 100 GB...
The best advice anyone can give: make a backup when messing with partitions, operating system installing or re-installing. Or in general: make sure files that are irreplaceable are on a restorable backup.
For this purpose my files are on a different partition (/discworld) I mount (and do not format) during a fresh install and I edit
$ more .config/user-dirs.dirs
# This file is written by xdg-user-dirs-update
# If you want to change or add directories, just edit the line you're
# interested in. All local changes will be retained on the next run
# Format is XDG_xxx_DIR="$HOME/yyy", where yyy is a shell-escaped
# homedir-relative path, or XDG_xxx_DIR="/yyy", where /yyy is an
# absolute path. No other format is supported.
#
XDG_DESKTOP_DIR="/discworld/Desktop"
XDG_DOWNLOAD_DIR="$HOME/Downloads"
XDG_TEMPLATES_DIR="$HOME/"
XDG_PUBLICSHARE_DIR="$HOME/"
XDG_DOCUMENTS_DIR="$HOME/Documents"
XDG_MUSIC_DIR="$HOME/"
XDG_PICTURES_DIR="$HOME/Pictures"
XDG_VIDEOS_DIR="$HOME/"
So my Desktop points to that directory (it holds all my video's I tend to watch).
Currently the method for OFFICIAL releases when you need to re-install another OS is to NOT format /home/ if on a different partition or to choose the "preserve my /home" when installing another OS. That will work but you might need to clean up your /home afterwards due to config settings not used in the new OS.
The 1st and 2nd option in the image below have a comment stating that the current personal documents will be kept and not removed. Where the 2nd option will end up with a dual boot of 2 operating systems. This would be my choice: install the 2nd OS next to the 1st one, move files over to the new one and when happy delete the 1st OS. That is always possible with any 2 operating systems:
More info on the wiki.
But your assumption is most likely wrong: Ubuntu will not be slower or less stable. If you want a stable Ubuntu stick with the LTS and skip the in-between releases. 14.04 was rock solid for me.