To learn a new active pet skill, or to upgrade an old one, you need to stable your regular pet, go out and tame a pet that knows the skill you want, and fight with it until you get a text message that you have learned the new skill. Sometimes this happens immediately, sometimes only after several battles.
Once you have learned the new skill, right-click that animal's picture and ABANDON it, then get your regular pet back out of the stable and use the slingshot Training icon under the General tab of your spellbook (easy to miss because it is not under the Beast Master section), to teach it the new skill.
Many hunters believe you can learn a new skill faster if you melee in close proximity to the animal you are learning from. I also feed the pet until it is "green-happy" though it is unknown whether that also helps. Growl and all of the passive skills (armor, stamina, resists) are learned from your pet trainer in town.
If you have troubles learning a new skill from a wild pet, here are my suggestions:
Recheck the lists in the links above to make sure the pet actually knows this skill.
Turn off any other skill on its bar besides the one you want to learn, including Growl. You want the skill to go off as often as possible so use all focus for that skill only.
Have it fight solo against very easy mobs. Stand directly next to it and heal if necessary, do not fight the mob yourself even if you are attacked by it.
Continually manually spam the skill button for that skill (Bite, Claw, etc)
Some skills such as Prowl and Howl can be learned outside of combat, just find a quiet place to sit with your pet and keep spamming that skill.