Part 1 - Attack, Follow and Stay
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 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***
Single target DPS:
Cat: Rake (and Prowl)
Devilsaur: Monstrous Bite
Dragonhawk: Fire Breath
Raptor: Savage Rend
Scorpid: Scorpid Poison
Spirit Beast: Spirit Strike (and Prowl)
Wind Serpent: Lightning Breath
Moth: Serenity Dust
Wolf: Furious Howl
All of the above are pretty standard single-target DPS spells except for Furious Howl, Serenity Dust and the bonus ability Cats and Spirit Beasts get, Prowl. On the whole, these abilities (apart from Prowl) can be left on autocast, although since some of them are DoTs you may need to be careful if you want to use crowd control.
For Prowl, I personally have a macro on my player bar:
....since I like to use my cat for camping nodes in battlegrounds.
Crocolisk: Bad Attitude
Sporebat: Spore Cloud
Swipe and Spore Cloud can both be left on autocast, as their cooldown is quite short. However, keep in mind that Swipe only affects mobs in front of the Bear, and Spore Cloud acts like a Paladin's Consecration - it will only affect the area where it's dropped.
For Bad Attitude, I'd recommend pulling this onto the pet bar and leaving it off autocast, as it's a fairly long cooldown.
Bat: Sonic Blast
Bird of Prey: Snatch
Chimaera: Froststorm Breath
Core Hound: Lava Breath
Hyena: Tendon Rip
Nether Ray: Nether Shock
Serpent: Poison Spit
Silithid: Venom Web Spray
Worm: Acid Spit
As these stuns, snares, debuffs and DoTs are all situational and important to cast at the right time, I'd recommend pulling them onto the pet bar and leaving autocast off.
In some cases (Sting, Poison Spit, Lava Breath and Froststorm Breath, for instance), the spells are on fairly short cooldowns, so you may be able to get away with leaving them on autocast. However, you still run a risk of not catching your opponent out if your pet runs out of Focus or the autocast is mis-timed and wasted.
Carrion Bird: Demoralizing Screech
Tallstrider: Dust Cloud
Turtle: Shell Shield
Demoralizing Screech can be kept on autocast, as it's a fairly short cooldown.
Dust Cloud, Shell Shield and Warp are all situational and should be pulled onto the pet bar to use manually.
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!