What is the safe way to update to latest version? I am taking over a Drupal project that's running on Drupal 8.8.5.
Can I upgrade to Drupal 9.1.5, or do I need to upgrade for each version that's been missed?
In theory you should update to the latest recommended version of your current site's minor release. So in this case you should update to version 8.9.13 then you should be good upgrade. Remember to also make sure you do the same for all contributed modules and themes.
You might find using the Upgrade Status module helpful before upgrading.
As with everything you do, make sure you create a full backup of the sites files and database/s before attempting any updates.