For the duration of the fight, creatures in the immediate area will not be visible to you (but they may still attack you if they wander too close). Keep in mind that other players will still be able to see you and attack you if you are eligible for player vs. player (PVP). Other players will also be able to view your pet battle in real time, including the pets involved as well as their health as the match progresses.
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.
To actually get access to the beta, which has already started and continue to add more players, you need to sign up via your Blizzard account management page. Under Games & Subscriptions, scroll down to Beta Access and visit the Beta Profile Settings page. Once there, you’ll see a grid of available betas you can opt into. Once you’ve checked WoW Classic, hit Update Preferences and you’re set.
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.