What are tricks?
Tricks are cool, short animations that your pets can learn and perform. All pets have 6 different tricks to learn, and many have personalized tricks to their animal type. The images below show a Strawberry Shortcake Dragon at different stages to show what tricks look like before and after unlocking.
Before:
After:
How do I teach my pet tricks?
Head to the School and meet the resident Aussie Pet Trainer Shane! By talking to him you'll receive instruction to ask him to teach your pet new tricks. 
I didn't unlock all my pets tricks! What do I do?
Pet Tricks unlock per the pet's age, so to uncover all 6 of the tricks, you will need to have fully aged your pets. Complete some more needs to get your pet to full grown and return to Pet Trainer Shane to obtain the remaining tricks.