Upgrading procedures and what essential directories to copy over?
As Kevin has stated in the comment, the "official" procedure should be found here, and it's ever evolving. I don't believe any safe automated way currently exists 100% of the time (such as with Drush/Drupal Console/Composer/etc.)

In short, upgrading D8 sucks. At least as far as I can tell. If you plan on using composer with it, you can't rely on "drush up", lest you want it all to get wiped, etc. (Actually, I'm kind of confused as to why you're not using "drush up" on your D7 sites, since you clearly have drush installed.)

In short, the process is still very similar to what you're already doing with D7. The file structure has changed in D8 though, and you need to be aware of things like modules/themes/libraries being in the root directory, rather than the sites/ directory. All that is covered in the documentation.