So I thought I'd go ahead and do a write up for the Beasts of Fable (BoF) quests. This is more about how I go about doing them than the absolute best method because there's more than one way to handle these. They had a huge boost in the last patch which makes them less trivial than before.
Why do you want to do these quests? Well in addition to the super cute panda pets, the gossip on Wowhead says that these have a much better chance of getting you an upgrade stone than the Spirits or Tamer bags, which is cool! Let's go through them shall we?
Like most of the BoFs that heal, Nitun can be a real pain. He's a Critter that does massive amounts of beast damage, and if he kills your pet with Ravage, he gets back 30% of his health.
Rake reduces your next attack by 50%, Prowl increases his next attack by 150% (and lasts up to two turns), and Ravage lasts two turns, hits extra hard and heals him when he kills your pet with it. He also has massive amounts of power, is pretty fast, and has a ton of health.
Like many of the BoFs, an easy way to take down Nitun is through abuse of Shattered Defenses. Since he is a Critter that does all Beast damage, a good counter is a team of Flyers that do Beast damage. So you can open with either a Flock pet or a Wolf/Fox pet to buff your team and debuff Nitun. You may choose to let the first pet get eaten, as this times it so your next pet gets two full turns with the debuff up, but you should swap your other pets before they're dead as he heals himself when he kills them.
Ka'wi the Gorger
Ka'wi the Gorger is a bit easier than most of the BoFs, mainly because there's no heals to worry about. Still, she's a big caterpillar thing that spits goo and shoots moth balls at you. Not exactly a fun time.
The good news is that Ka'wi is another Critter. The bad news is she does all kinds of Flying damage in addition to Critter damage, so you can't just pick on her with one pet type. Also, because of Sticky Goo you can't do constant swaps and try and abuse Shattered Defenses that way.
The pet of the match for this is a fast (>281 speed) Squirrel or Woodchuck. You start with a bleed debuff. Crouch, and then Stampede in order to put up the Shattered Defenses debuff. Rinse, repeat, and Ka'wi will go down quickly.
For extra points, you can bring in a pet that does Maul and finish her off, but by then she should be pretty much dead so it won't matter.
Of all the BoFs, Dos-Ryga probably gives people the most grief. He does a ton of damage, both Aquatic and Dragonkin, and has a ton of healing.
You are going to need three good pets for this one. One of them (like a flock pet or a fox) should put on Shattered Defenses and the other two can hit Dos-Ryga with Reckless Strike for a huge chunk of health. The sad thing is, that he'll heal much of it back so it may take two pets with Reckless Strike to really kill him.
A Gilnean Raven can help, but you really need Shattered Defenses + 1-2 big hits to keep him from out healing your attacks. Call Darkness can help make up for it if you miss, however, and the Raven does a large amount of damage on its own.
Kafi is about typical for a Beast of Fable. Which means he's stronger and faster than your whole team put together!
Tons of Beast damage, a Beast damage with a speed buff, and some extra Beast damage. While you can try and put a Flyer on him to reduce the damage, your best bet it to just burn him down quickly.
So the plan is to just bring your hardest hitting Mechanicals and beat him into submission. If you want to get fancy, bring a pet with Shattered Defenses and make your Tonk (or whatever) hit him that much harder. As long as you have enough Mechanicals in your stable, flying out there is the hardest part.
There are some tough pets here, but they are balanced by some relatively easy ones. These fights are more about choosing the right pets than any kind of tactics. You can do a bit of fancy stuff by choosing a Bleed + Maul team, or a Shattered Defenses + Reckless Strike team, to build a combo, it's less important than picking pets that do extra damage against the BoF you're fighting. Fortunately, it doesn't take long to do these and you can get some cute pets or a nice upgrade stone as a reward.