Just to clarify, these are not pets like hunters have. These will not help you battle. These are vanity pets, and the only battling they do is against other vanity pets, either wild ones or ones that other players have in a pet pvp battle.
Ask the guards in either Stormwind or Orgrimmar where the pet battle trainer is. They'll mark it on your map for you. Also, check your pet tab, it grouped with mounts, the access button has a horse's head at the bottom, from memory I would say it's to the right of the guild button and to the left of the logout button.
Almost everyone should have at least one pet available, either from logging in during a wow aniversiary or something. Oh!, if you don't, then talking to the battle trainer, she'll give you one that is class specific. From there it's just a matter of expanding your collection and increasing your pets lvl. When you battle a wild pet, and you get him to something like 20% health, you'll have to option to throw a cage at him, when this lands eventually, you'll have captured your new pet and added him to your collection.
There are a ton of guides out there, hell, this website has a very nice one. Let me link it. http://www.warcraftpets.com/wow-pet-battles/
Read that, that's what I read to get started.