Note that you can only learn 4 active abilities at any one time per pet. As you level you’ll be able to train your pet with ever increasing levels of these skills granting more stats as you go. Most of your default pet training will come from the pet trainer located near the hunter trainer who comes with a specific set of predefined skills your pet can learn.
This is quickly followed by three subsequent quests that include using a taming rod to tame three different types of creatures. Taming your first pet comes at the end of a level 10 quest called The Hunter’s Path that you get from your hunter trainer.