Becoming a Master
A new quest will pop up once your first pet reaches level 3. It will send you to either Audrey Burnhep in Stormwind or Varzok in Orgrimmar. These trainers will offer a quest chain to battle specific Pet Tamer NPCs on each continent. Defeating each of these NPCs will reward a nice chunk of experience for your pets, as well a Sack of Pet Supplies (which has the chance to drop the Porcupette companion, among the always-useful pet bandages).
Pet Tamer NPC
There are two parts of the quest chain for each continent. The first part involves defeating numerous Master Pet Tamers, and the second requires you to triumph over the Grand Master Tamer. After you've bested the Grand Master Tamer, dailies from each Tamer will unlock for that continent. All the dailies continue to reward experience for your companions, however only the Grand Master daily quest will reward the Sack of Pet Supplies.
Keep in mind that these quests are optional, and are not necessary to continue leveling and capturing companions. They are also account-wide, so if you complete the quest chain on one character, you will not be able to complete it again on another. However, any progress that you've made will be saved and you can pick up where you left off in the quest chain on another character. The dailies that you've unlocked will also be accessible by all your toons.
For more information on each Pet Tamer and the complete quest chain, take a look at this post in our forums.