I'm using Drupal 7.10 which is an outdated version. Is it safe to just upgrade to the latest version or should I upgrade version by version from 7.11 until the latest version 7.21?
You can upgrade directly to the new version as it contains all the previous changes as well. But don't forget to follow the backup instructions...
just go to your command prompt , switch to your site folder and use the command "drush pm-update drupal" this ll help you out ..:) in an easy way..
happy coading :)