An alt (low level or no) in a capital city is an effective way to not only cheaply increase your available bank space, but to be a simple savings and auctioneer account. This character can serve as your bank, an auctioneer, bag-space creator and a time saver. Get one. (There may or may not be Item Recovery issues with characters below level 10, in cases where one's account is hacked.)
Although the Battle Pet Trainer will only offer to teach you your race-specific companion, it is possible to add all of these pets to your collection. You can either find them in your faction’s auction house or you can create a character of the race whose pet you would like to learn. Using your new character, you can seek out any battle pet trainer in major cities. After you learn the pet, you can delete the character; the pet you learned will not be removed from your collection.