Typically, when starting out you want to level a single pet to level 25 as quickly as possible, and ideally one that can solo trios of wild pets between 20 and 24 by itself, preferably from as early on as possible. So you're probably looking for something with at least one heal. Dunno what you have access to, but my first pet to 25 was Sinister Squashling (planted is a great heal), then my next 3 were Hopling (humanoid - so it effectively heals each turn on account of its family ability), Spawn of Onyxia (that dragon heal it has is very potent) and Toothy (consume is an aggressive heal on a short CD; devour heals for a *lot*).
If you don't have anything decent already in your arsenal (I've heard good things about Magical Crawdad, but many other crustaceans have similar move sets), then I'd recommend going with a Spider. You can find them at low levels, they have a heal that has good synergy with their second ability (webbing), and are a beast so get bonus damage vs critters. Also, make sure you have an addon or macro for telling the quality of wild pets - you really only want to battle with rares where you can help it (and you can usually always help it).
After that, grind the heck out of that first pet, doing tamers whenever it's strong enough to pretty much solo them. Tamers offer more xp than wild pets, and generally it's the fastest way to level that first pet. Once it starts getting close to lv 20 you want to find some wild pets in Northrend+ zones that you're comfortable levelling on, preferably close to a stable master, and then just do that non-stop until 25 (given that it's likely your only high level pet, you'll only want to start when you can comfortably do a battle without your pet dying - if you can't, go for more tamers or lower level groups of pets).
Once you have that first 25, that's when things get a lot easier. With any new pets, you put them in the lead, and have them perform an ability in the first round. After that, assuming they survived (hint - if they're not surviving start with lower level groups until they get a health buffer), swap to your 25, and clean up the pets. You'll get a lot of xp this one, and against 22-24 wild pets it takes around 6 battles of this style to go from 1 to 10. All in all I think it takes about 60-80 of this style of battle to go from 1 to 25 - which isn't bad at all. In about 2 hours of battling you can get any pet to 25 - what's not to love! :-P