In my ongoing effort to keep my customers sites secure and all modules update for the best possible webpage response, I have recently discovered how to update the core when needed.
Messages will appear at the top of most of the admin pages if a new core or security update has been released. Most likely you will also receive an email to the administrator of the site.
Here are the steps to safely update your core:
1. Backup your site. (Install the “backup and migrate” module from https://www.drupal.org/project/backup_migrate)
2. Put your site in maintenance mode: Configure –> Maintenance Mode
3. (check) Put site into Maintenance Mode
4. Download the latest version of Drupal: Drupal.org –> Download & Extensions –> Download Drupal 7.35 (or whatever is the newest)
5. Extract the file on your machine and ftp all the files & directories ACCEPT SITES over the existing drupal files in your drupal directory
6. Next goto yourdomain.com/update.php –> Click continue –> Run updates
7. Next clear your cache Configuration –> Performance –> Clear all cache
8. Take your site out of maintenance mode: Configure –> Maintenance Mode
9. (uncheck) Put site into Maintenance Mode