That depends on what you mean.
If you want to tame the beasts of the world, you need to be a Hunter.
If you want vanity pets, then there are various ways you can acquire one. Some are holiday pets only, like a tiny reigndeer or snowman. Others like the Dragonhawk Hatchlings are easily bought by vendors. And some are craftable by engineers like the Bombling and Mechanical Squirrel. The Mechanical Chicken you can get from a quest, and some vendors are rare spawns like Lil' Timmy who sells a white kitten in Stormwind.