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.

The Beastlore ability (learned from your Hunter trainer at level 24) is indispensible for learning new pet abilities from taming.  You cast it on the target beast and it tells you what special abilities, if anything, the animal teaches.  That way you don't waste time training an animal that doesn't give you the ability you want.  Personally, I think this should be taught at lower levels, and at much lower cost (I think it costs around 60 silver).  It also tells you what food the beast eats, so you don't end up training a bat that eats only fruit when all you have in your bags are meat products.

I understand that this can be difficult for new players, so lets see if this helps.
Most animals that you make your pet will most likely not have any abilities other than basic attack. You have to learn other abilities by learning them from other animals. How do you do that, you ask? Like this:
1. Place your current pet into a stable
2. Go to http://petopia.brashendeavors.net
3. Click on the type of pet you currently have and want to train. For an example, lets pick a cat.
4. When you click on cat, the next page that comes up shows you various information. TAKE NOTE OF THE AREA THAT SAYS "LEARNS". These are the abilities you can teach the cat. Lets say, for this example you want to learn bite. Soo.....
5. You click on bite
6. This page shows you what animals you can learn the various ranks for bite. As an example, lets say you are close to Mulgore. The animal that will teach you bite rank 1 in Mulgore is the Prairie Wolf.
7. See how in parenthesis it says "5-6"? That is the level of the Prairie Wolf. You have to be at or higher than that level to tame it.  In other words, you cannot tame a level 6 Praire Wolf if you are level 5.
8. According to the list you can also learn Bite rank 1 from a Prairie Stalker in Mulgore.
9. These are the exact names of the animals, so you cannot learn Bite rank 1 from an ELDER PRAIRIE STALKER. The name of the creature is a PRAIRIE STALKER.
10. Go out, find the Prairie Wolf (for this example), and tame it.
11. Feed it, so it wont turn on you.
12. Fight with it and watch your chat screen.
13. Soon, you will see a message that says "You have learned a new ability. Bite Rank 1".
14. Abandon this pet by right clicking on his picture on selecting abandon.
15. Go back to the stable and get your cat.
16. Click on the training icon that looks like a slingshot, select Bite rank 1 and train your cat (providing you have enough training points available).
17. There ya go, ya got Bite rank 1. Congratultions.
18. Pick another abilitie, like Claw rank 1, rinse and repeat.

You dont need to get bite 1, 2 , or 3 to get 4 or any of these

You only need the pet and like i said in my other post, if you dont get it right away just wait, kill a lil with it , it tooked me not even 10mins to have it, its will say the u have learned bite rank x (x is for the num like 1 2 3 ..)