Best way I've found:
1. Get a Flayer Youngling to 25. Should be easy; he destroys critters, which are about 50% of the wild pets you'll fight.
2. Go to the Veiled Stairs (the area near the BMAH) and start the fight with the pet you want to level.
3. Attack once, then switch to The Flyaer Youngling.
4. Mop up the critters, enjoy the thousands of experience your lowbie pet gets.
The Flayer Youngling is a hard counter to critters, and there are ONLY critters in this area. If you get good at timing your Reflections, you can go 3-4 fights without needing to heal... and even when you do heal, you have a stable master right nearby. I can level a pet from 10 to 25 in about 30 minutes here, with no risk at all.
The hardest part of this is getting a Flayer Youngling, honestly (especially a good breed and rare quality). I think the Wolvar Pup works too, as they are very similar.