Another thing you can do, just for general leveling and not with the tamers is go for a team that's really good at self-sustainability. You have the Feline Familiar, one of THE best pets for self-sustained leveling of other pets. Get that one to 25 ASAP and then you can use it to fight any flying or critters teams in the Vale of Eternal Blossoms or the Valley of the Four Winds. Use Onyx Bite as your main attack, keep Stoneskin up at all times, and use devour when the enemy is almost dead. You can pretty much battle non-stop with that 1 pet.
Keep in mind, your carry pet needs to be able to take 1 hit if you want to start it in the battle, so it needs a couple hundred health (level 5 or 6 is usually enough IIRC). What I do is I get a pet with a 100% stun move (like most bears) and to level up from 1 I start with the bear (fighting non-critters, this is critical!) and open with Bash, swap in the level 1, then immediately swap in the Feline Familiar to finish off the enemy team.
Since you are just starting out, I'd recommend levelling your Feline Familiar and Cheetah Cub (nice S/S so his Devour will go off first most of the time). Get yourself a bear at some point when you are ready to level other pets, and in the meantime take a look at the 5.4 guide for a list of pets you may want to collect to get you ready for the Pandaria Tamers, and also the posts on the Celestial Tournament for those pets, too.