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.
In player vs. player designated zones, you will still be viewable and attackable by players, but you are not left completely defenseless. While engaged in a Pet Battle, any attack from another player will break you out of the battle. You will receive a 3 second damage shield that will absorb 50% of incoming attacks. The same exact wild pet that you were previously battling will respawn after you come out of combat with the other player. However, if you defeated the primary pet in battle prior to leaving the match, the pet and its team will not respawn.
With the Classic beta now out it seems every related article somehow manages to spark the eternal war of "Vanilla was the best WoW sucks now" and "lol nostalgia goggles, Vanilla sucked, enjoy your two weeks of Classic". I have to say, even though I understand the principles behind the battle and the reasons people behave and talk this way... I actually REALLY don't get it on a deeper level.