Old World Pets

Here are the locations of all of the Old World Pets.

World of Warcraft's tamable pets

Cats: Are purchased from a 'Crazy Cat Lady' just east of Stormwind in Elwynn Forest (44, 53).
Orange Tabby
Silver Tabby
Cornish Rex

White Kitten: Sold by Timmy in Stormwind. He isn't really rare but despawns rather quickly after selling just 1 kitten. If he just doesn't offer you anything for sale then he has already sold his 1 kitten. Try again later.

Owls: Owls are purchased in Darnassus.
Hawk Owl
Great Horned Owl

Snowshoe Rabbit: Sold in Dun Morogh

Snakes: Snakes are sold from a wandering vendor in the Valley of Spirits in Orgrimmar.
Black Kingsnake
Crimson Snake
Brown Snake

Prarie Dog: Sold in Thunder Bluff.

Cockroach: Sold under the bank in UnderCity.

Ancona Chicken: Sold in the Shimmering Flats in Thousand Needles.

Prairie Chicken: Short quest in Westfall from the chickens around Saldean's farm.

Parrots: Sold in Booty Bay.

Macaws: These parrots are rare drops.

Hyacinth Macaw: found in Stranglethorn Vale

Green Winged Macaw: found in The Deadmines

Mechanical Squirrel and Yeti: Made with Engineering (usable by anyone).

Mechanical Chicken: Quest reward after completing the three beacon quests.

Exotic Pets: These pets are quest rewards from a guy in the Burning Steppes. They can be soloed with the right class but usually require a group in LBRS.

Smolderweb Hatchling (spider)
Worg (dog)

Dragons whelps: Dragon whelps are rare drops from dragon whelplings.

Tiny Crimson Whelpling: found in Wetlands

Tiny Black Whelpling: found in Badlands

Tiny Emerald Whelpling: found in Swamp of Sorrows (I think)

The Disgusting Oozling is a rare drop from various Oozes around the world. It is unusual due to it aura. It turns the owner green and even lowers some resistances.

Collector's Edition Pets: After loading up your character, you find a present in your inventory that you give to an NPC in exchange for one of these pets.
Mini-Diablo (Lord of Destruction)
Nether Drake (Burning Crusade Expansion Pet)

The was a Murloc pet named "Murky" given out at one of the Blizzard Festivals. You had to redeem a coupon in Ironforge to obtain him.