UPDATE #2: According to Mumper they should have the names you've assigned them in the Pet Journal. It requires your garrison to be reloaded, though. Time to rename all my pets and the relog to double check!
Also, the pet battle daily is NOT account wide (for now). I was able to create a second character and pick up the quest again. This might change so we'll have to wait and see what happens once WoD goes live.
Now that Blizzard has opened up a level 100 PVE server, it's fairly quick and easy to make a premade toon and check out the Pet Menagerie in your garrison. I covered my initial findings in the menagerie in this post, but now that I have full access, I've tested a few more things.
One neat little cosmetic perk is that once your menagerie is fully built, four pets from your "favorited" list will appear nearby. They currently don't really do anything other than sit/hover there and look adorable, but it's a nice little touch. I just wish I could coax them to follow me around while in my garrison; I've always wanted a pet army following me! :P
Completing the Menagerie
Although the plot will be unlocked, you'll need to finish building the menagerie itself. To do this, Lio the Lioness will have a quest for you to complete, Pet Versus Pet. The objectives are straight forward, defeat three pets that can be found around your garrison.
The three pets you'll need to defeat include Gnawface, Carrotus Maximus, and Gorefu. All of the pets are legendary, and as such have the Boss buff.
In the case of Gnawface, who also uses Crouch (reduces damage by 50% for three rounds), the buff can be nasty. With Crouch up, Gnawface will take zero damage from unbuffed attacks. (UPDATE: LeCraft noted that this is a bug and Crouch will be replaced with Comeback.) Much like the other legendary pets that we currently find in Pandaria, the most effective strategy against Gnawface and his allies is to stack damage debuffs on them and then go to town.
Against Gnawface, I brought out my trusty Zandalari Raptor trio (thanks to @gloriaboboria's suggestion!). Black Claw and then Hunting Party did the trick. With good RNG, I didn't even need my second or third pet. It helped that I had them there though, just in case Gnawface scored some lucky crits.
Next I faced Carrotus Maximus and used the proven Unborn Val'kyr, Pandaren Water Spirit and Chrominius team. Unholy Ascension and then I swapped to my Water Spirit to put down Geyser and Whirlpool before switching to Chrominius. Howl, Surge of Power and win!
Last but not least, Gorefu. Quite frankly, the previously mentioned teams would probably work just fine against this pet, but I wanted to try and create completely separate teams to use against each pet. On top of that, I had a desire to make a team that utilized mainly strong attacks. Maybe that goal was silly and unrealistic of me, but in the end it worked out ok. Just ok lol.
I ended up using Ji-kun Hatchling (2,2,1), Darkmoon Hatchling (1,2,1), and any chicken (1,1,2). Started with my Ji-Kun Hatchling, Acidic Goo and then Flock until my pet died. Swapped to my Darkmoon Hatchling, Hawk Eye and then Flock again until it died. At that point Gorefu had low health and an increased damage debuff on it, so my chicken was able to use Squawk and then Peck to finish him off.
It's probably not the best strategy, but like I said, I wanted 3 different teams, each utilizing "strong" attacks against each opponent. I probably won't stick with these teams once more testing's been done and the pet theorycrafters come up with better ideas, but these teams satisfied my desires for now. :P
Completing each objective will cause the NPCs to despawn, and turning in the completed quest will cause them to disappear altogether. So if you're looking to fight the three pets multiple times, abandon the quest each time and pick it back up. I didn't test whether or not they yield any pet experience, but I imagine that it's not much if they do (same as the Pandaria legendaries).
If you want to start doing the garrison pet battle daily, though, you'll need to turn in the quest. Lio will then have a daily ready for you. It's appropriately called "Pet Battle Roundup" (name of Liopleurodonic's pet battle blog!), and the objective is to simply defeat whichever trainer shows up at your garrison that day.
Completing the daily rewarded me with four Beast leveling battle-stones. I'm not sure if these stones will change daily, or if they will always be from the Beast family. Even if you turn in the daily, you can still challenge the trainer multiple times, just for funsies or perhaps testing purposes. There's really no benefit, though, unless these battles award pet experience. Again, it's something I haven't tested, so I can't say if they do or not.
As with all things beta, there are bound to be changes and updates. Lio the Lioness is no exception.
This vendor received an update since my last blog post on the menagerie. She no longer sells family-specific leveling battle-stones, and instead has Flawless versions (generic, can use on any type of pet but still only grants 1 level per use). On top of that, she also has a stock of Marked Flawless Battle-Stones, that you can trade for by giving her 3 stones from the same family (same as the battle-stone vendor in Stormwind and Orgrimmar).
Her vanity selection has also been cut slightly, and she doesn't have an endless supply of Perky Pug costumes anymore. There is only a limited supply - I guess these hats are hard to import from the Timeless Isle? :P
She still only accepts Pet Charms as currency for many of the items. Unfortunately, it's still unknown where these charms come from. I thought they might be linked to the daily in some way, perhaps the reward for completing it each day, but there was no such reward. So it's still a mystery.