What does a premium account do? 

Copied from the FAQ:

1 Premium users can set Triggers.

2 Premium users see maneuvers that are currently selected by your opponents at the Battlefield. 

3 Premium users get Inbreeding Warning when trying to breed pets that are close relatives.

4 Normal users can perform searches only among pets which are being traded. Premium users can search among all pets. This can be useful if you're looking for a pet with special traits and want to buy it from another Keeper who didn't put this pet up for trade.

5 Premium users can search among their own pets.

6 Normal users only see a pet's mother and father. Premium users have access to full genealogy.

7 Premium users can instantly leave the island from any point of the map. Normal users need to move to the portal tile first.

8 Premium users can send a group of pets to Arena and cancel all battle applications for a group with one click.

9 Premium users can see their pets ranked for the current exhibition before applying them to the exhibition (valid only for exhibitions that don't have random parameters involved).

 You can see the full FAQ by going to your Help tab at the top right of the page. Hope this helps!