Upgrading procedures and what essential directories to copy over? When I upgrade my Drupal 7 core I do the following process:


Download and unzip d7 release as d7_upgrade
Delete the sites folder in the d7_upgrade
Move the sites folder from my original site into d7_upgrade
Rename the old D7 folder to d7_old
Rename the d7_upgrade to original folder name
drush updb


Majority of the non-core files reside in /sites for D7. 

I know D8 has a different structure, but is the same upgrade process going to be the same? I know I would have to move over /themes /modules /sites

Is there anything else I need to move over? Or is there a better method of doing this? I want to upgrade to D8.3 beta to test on my site.
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.