I hav heard of pet stables but not sure how they work.
Stable masters are generally found next to inns, and in Dolanaar it's Seriadne (http://www.wowhead.com/npc=10051). Just speak with her to store your current cat, then go out and tame the one you want. You can also just right-click on the cat's portrait and select "Abandon," but that will permanently release the cat, and it'll never be available again; if you want another white tiger, you'd have to tame a new one (and there happen to only be two of those kind in the game, a level 29 in Stranglethorn and a level 70 rare in Outland). That option is good for when you're running between the stable master and the location of the pet you want, and just need something temporary along the way.
If you haven't already found it, WoW-Petopia.com is a good site for learning where to find more hunter pets.
Battle pets are completely different. They're just little vanity pets that can fight each other, much like Pokemon.
And when you reach Darnassus, there are a few vendors near the main gate, off to the left. One of them sells little owls. The Pet Battle trainers also sell a basic vanity pet, so if you decide to try Pet Battles, you can start with one right away.