I'm pretty new to civiCRM and it feels like I'm missing something obvious. The tags that are in tag sets don't show up in the Find Contacts search. They show up in Advanced Search just fine. Is this just how tag sets work or is there a way to make them show up in both search options? I'm using Civi 4.6.10 in Wordpress.
Matthew, welcome to the CiviCRM community.
Have a look at http://book.civicrm.org/user/organising-your-data/groups-and-tags/ - there's a section on using Tag Sets.
It does say "adding them creates an additional search field in the Basic Criteria section of the Advanced Search" indicating that they are not available in the Find Contacts search.