Where does drupal store all its files? I would like to use FTP to copy my current site to a new server. I obviously know that the bulk of my files will be stored in /var/www/ and the folders inside. I also exported my databases from phpMyAdmin to a localhost.sql file.

Is there any other place I need to pull files from? Is that it for the database?
You should be able to transfer the site wholesale, without too much fuss. After transferring, you may need to alter some paths (eg admin/config/media/file-system in D7) if the sites directory has changed paths relative to your Drupal codebase. Odds are you won't need to, though.

For a simple site, the layout will be something like this -

/var/www/index.php 
         modules/                  } don't install modules here
         themes/                   } nor themes here
         includes/
         [many other Drupal core files]
         sites/
               default/            } all installed code
                       files/      } lives in the
                       modules/    } site directory
                       themes/     } and its children


For more complex setups you may need to understand multisite layouts, but for now hopefully this answers your question.