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.
Each pet has 3 abilities it can use during a battle, but a total of 6 to choose from. They will also have a total of 3 active spell slots that show which abilities can be used during a Pet Battle. The first spell slot will already be unlocked, but the second and third will only become available once the companion has reached a certain level. Other spells will also unlock after reaching higher levels.
If your pet's health gets too low or its attacks aren't very effective against a certain combatant, you can always switch out your current pet with another companion on your team. Only pets that are considered active (i.e., those in your three Battle Pet Slots), may be switched in and out during the battle. Once the match is over, you can always change up your active pet team, which is useful if one of your pets is low on health or dead.