1733 pets
The Pandaren elemental spirits that were recently datamined from the 5.1 PTR are listed on Wowhead as quest rewards for a new Tamer quest, Pandaren Spirit Tamer.There's also a daily quest that shares the same name.
According to Mumper, lead content designer, this new Spirit Tamer quest will only be available after you have completed the main Pet Battles quest chain (currently ends with the quest Zen Master Aki). [source]
These quests are flagged as account-wide, which means you may only complete them once per account. However, your progress is saved, so after unlocking the daily quests on one character, you can pick it up and complete it on another toon.
Collecting All Four
The main quest offers you a choice of only one out of the four pets, but don't worry! You'll be able to obtain all four Pandaren elemental spirits.
Mumper confirmed on Twitter that although you can only choose one pet from the initial quest, each spirit has its own daily bag reward that has the chance to drop the corresponding elemental spirit. [source]
So get your pet team ready, more Tamer quests and dailies will be coming in the near future!
