Play the games in the pet pavilion. At first it seemed dumb to me but it's no different from taking care of your home. Water the plants; feed the dog; etc. I like Grumpy Gobblers but there's something for everyone. Your pet is from a school just like you. If you have multiple wizards, all pet snacks can go into the Shared Bank. That way when you've picked up dropped snacks after duels and they're not what your pet likes you can pick out your animal's preferences that your other wizards have collected. Feed your pet from his/her school. They grow faster and do much better for you. Also compliment them when they do something for you. My hound is Spritely and I always tell him "Good Boy" when he heals me in battle and after games.
when ya get access to the pet pavilion ( at level 7 or so merle ambros will give you a quest where ya must deliver a little friend to the ring master of the pet pavilion ) he will show you all there is and to train your pets you must play pet games ( or if you love playing tower defence playing games ) you can play grub guardian on kifreegames.com to train your pet with your ingame snack inventory while playing grub guardian ( wizard101 tower defence like game )
Go to Pet Pavilion (unlocks at level 7) play a game, then feed your pet a snack. Eventually it will level up and gain a derby ability and a talent that should help you in battles...Hope i helped.