Part 1 - Attack, Follow and Move To
Part 2 - Aggressive, Defensive and Passive
Part 3 - General pet abilities
Each "species" of pet has a unique ability. Some of these are tailored for DPS, others for PvP, others for tanking... and some are just random bonus abilities that suit the pet but aren't as useful. Below, I've separated the various abilities into specific uses for comparison.
***Note: I am NOT going to argue the pros and cons of each pet here. There are plenty of other places where you can find out which pet is best for what purpose; this guide is about pet control only***
Raid buff or debuff replacements:
Bear: Demoralizing Roar (and Rest)
Devilsaur: Terrifying Roar (and Monstrous Bite)
Dragonhawk: Fire Breath
Hyena: Tendon Rip
Fox: Tailspin (and Play)
Raptor: Tear Armor
Ravager: RavageRhino: Stampede (and Horn Toss)
Serpent: Corrosive Spit
Shale Spider: Embrace of the Shale Spider (and Web Wrap)
Silithid: Qiraji Fortitude (and Venom Web Spray)
Spirit Beast: Roar of Courage (and Spirit Mend and Spirit Walk)
Sporebat: Spore Cloud
Tallstrider: Dust Cloud
Wind Serpent: Lightning Breath
Worm: Acid Spit (and Burrow Attack)Wolf: Furious Howl
Most of the straight buffs and debuffs can be left on autocast, as you will want these to be up at all times. However, if there are adds in particular fights that die quickly and your pet is required to switch targets frequently, it may be better to keep pet debuffs on long cooldowns (such as Fire Breath) off autocast so that it can be fired on the main target as soon as possible.
For the pets with extra abilities (mostly the exotics), some care is needed in using them.
Ancient Hysteria, the equivalent to Bloodlust/Heroism and Time Warp (and sharing a cooldown with these), can only be used once per fight, so you will need to determine exactly when this should be cast and be ready for it. Keep in mind that if your core hound is incapacitated or in the middle of casting another ability (eg. Bite), it may not cast and you'll need to spam the ability to make it work.
Utility spells (Spirit Mend, Horn Toss, Venom Web Spray, Web Wrap and Burrow Attack) should be off autocast. I recommend pulling these onto the pet bar to use as appropriate; Horn Toss, Venom Web Spray and Web Wrap to CC adds and Burrow Attack for AoE.
For Spirit Mend in particular, a macro may work better, as this ability needs to be cast on a friendly target. For example:
/cast [@mouseover,help] Spirit Mend(Exotic Ability)
...will cast Spirit Mend on your mouseover target.
For Prowl (and Spirit Walk as well, if I ever manage to tame a Spirit Beast), I personally have a macro on my player bar:
....since I like to use my cat for camping nodes in battlegrounds.
Play and Rest are bonus abilities, which I tend to leave in the spellbook and manually find when required.
Bat: Sonic Blast
Beetle: Harden Carapace
Bird of Prey: Snatch (and Trick)
Crocolisk: Ankle Crack
Chimaera: Frost Breath (and Froststorm Breath)
Dog: Lock Jaw
Monkey: Bad Manner
Moth: Serenity Dust
Nether Ray: Nether Shock
Scorpid: ClenchSpider: Web
Turtle: Shell Shield
Warpstalker: Time Warp
For almost all of these, I would recommend pulling them onto your pet bar and manually casting them as required. Frost Breath and Ankle Crack could possibly be left on autocast due to the short cooldown, but you really want the rest to be ready when you need it.
Trick is another bonus ability and as with the others I'd leave it in the spellbook and save your slots on the pet bar for something else.
Now that all the pet abilities have been covered, the fifth and final part of this guide will focus on how to organize the million and one pet spells onto 4 measly spell slots on your pet bar!