Well, there are two classes that you can be to get a pet, the Warlock and the Hunter. For the warlock your first pet will be attainable at around level 4. Then there are other pets that you get at level 10, Voidwalker, level 20, succubus and the level 30 pet, The Felhunter.
If you are a hunter, you get the Tame Beast skill at level 10, and you can tame any beast you wish. Make sure you pick up the quest for the Tame Beast ability at your Hunter trainer.
Hope I have helped, and good luck with it, whether you are a Hunter or a Warlock.