First off, I would strongly suggest getting a team (or more) to 25. Hitting all the trainers (you'll probably want to for the quests anyway) helps this along nicely.
I find it helpful to take whatever prospective levelers I have to a main city and do a few fights to get a handful of them to 5 or so real quick.
Take a pair of 25's and a lowbie - I find 5 a safe level to start at - to a pet-rich area of 23-35s. Southern Kun Lai is great I think, nearly all beast/critter types. Do the 1 round swap with the lowbie and burn through them with your 25s. You can get a pet from 5 to 15 in a few fights real quick that way. Usually takes maybe 30-60 mins to go from 5-25.
The trainers are great, and do provide a big xp bonus, but they're limited and scattered. So if you want to hit a couple before camping a spot that's probably great but it seems less time efficient to me to go around hitting them all than just parking somewhere and grinding.
Once you get those first few capped out though, it gets way easier. You'll get to know the areas and what types of enemies to expect so you can take out the 3-pack with 1 or 2 of your own easily freeing up a spot for uberlowbie xp padding.
I know the numbers *should* suggest otherwise, but I've tried leveling multiple pets at once and it just seems so much slower to me. It seems like I can get 3 pets from 5-25 quicker this way than, say, leveling a set of 3 15s to 25 together if that makes any sense.