Pets can learn bite, claw, cover, growl. All pets can learn growl and cower; most can learn bite, and some can learn claw.
Growl – available from pet trainers, can also be learnt from pets tamed at the appropriate level. Levels are: 1, 10, 20, etc.
Bite – Level 1, 8, 16, etc.
Claw – Level 1, 8, 16, etc.
Cower – 5, 15, 25, etc.
To get the highest available level of each ability when you start:
A starting Night Elf should tame the following: Webwood Silkspinner, Strigid Hunter, Nightsaber.
Then in Darkshore: a Giant Foreststrider or Moonstalker Sire (for Cower 2).
A starting dwarf should tame: Starving Winter Wolf, Ice Claw Bear (claw 1) or Mangeclaw (claw 2), Juvenile Snow Leopard. The only problem with the Starving Winter Wolf is that he I have only seen him on the wolf island in Iceflow Lake accompanied by two close friends.
To make best use of this, buy an extra stable slot (5s) for your main pet, and put him in it. Go off and tame the temporary pet, and fight with it until you see the message that you have learned the new skill. It works to kill non-experience mobs to learn the skills.
See Hunter Pet Abilities for further details.