The trainer will only teach you your race's
home pet. Only Worgen can purchase the Gilnean Raven from Battle Pet Trainers. Nelfs learn Great Horned Owls, Draenei learn the Blue Moth, Trolls learn the Black Kingsnake, Pandaren get the Jade Crane Chick, etc.
Regardless of which trainer you speak with, they'll only offer you your race's home pet. Base cost is 50 silver.
Note: deleting a character after it's trained his/her race's pet from the Battle Pet Trainer does not remove that particular pet from the pet journal.
So I guess if you don't have any real desire to play a certain race but would like their home pet, you could just create an alt, purchase the ability to battle and then the pet from the trainer (to buy the pet it says Pet Battle training required), and then delete the toon.
Update: Here's a list of all the races and their "home pets" that can be trained/purchased from a Battle Pet Trainer.Alliance
Draenei - Blue MothHorde
Dwarf - Snowshoe Rabbit
Gnome - Snowshoe Rabbit
Human - Orange Tabby Cat
Night Elf - Great Horned Owl
Worgen - Gilnean Raven
Blood Elf - Golden Dragonhawk Hatchling
Goblin - Shore Crab
Orc - Black Kingsnake
Tauren - Brown Prairie Dog
Troll - Black Kingsnake
Undead - Undercity Cockroach
Pandaren - Jade Crane Chick