When you want to train a pet, you head on over to the Pet Pavilion in Wizard City. You can get to it from the Commons. Play one of the games available, do your best , have fun and at the end of the game, if you have any, give your pet a pet snack. Repeat this multiple times.
Doing so will cause your pet to gain experience, therefore training them. Eventually they will level/rank up.
There is no way to train it fast.
You have to train it in pet pawvillion.
You have to have a pet
The pet store in wizard 101 is at Shopping District. Use your map at you journal. Or your compass.
you can't ride a pet on wizard 101 unless you buy one from the crown shop
The pet pavilion is connected to the Commons of Wizard City.
Some pets naturally have it or you can train it to get talents and it might get a sprite spell
Wizard City, Pet Pavillion
You train it in the Pet Hatchery by playing mini games and feeding it snacks. Once your pet is adult, it can be hatched with other pets to make hybrids.
-Summer Dragon (Pet Dragon)
there isn't a pet shop there
You have to buy it at the pet shop in the Shopping District.