One of the biggest features of drush is the ease of doing site upgrades. With security updates coming every few days, automating the process is essential.
As always, you need to cd to the site's document root folder.
Backups
Creating an archive of the site is the first step. The simplest approach is:
sudo drush archive-dump
which puts a tgz of both the document root and a sql dump into ~/drush-backups.