That's why I was interested when Sarial of The Uncommon Drop wrote in letting WoW Insider know about her goal to level a character in an unusual way. Her plan is to gain experience purely through the attainment of non-combat pets. There certainly are a plethora of vanity pets out there, and collecting them was once a passion I admit I spent a little too much bank space on.
As fascinating as this project might be, I'm not sure it can be done from level 1. The first non-combat pet in my recollection would be the one that drops from the Red Whelps in Wetlands after all. But Sarial will be grinding from level 40 to level 70 with her Blood Elf warlock, which seems to be a more reasonable range. There are certainly lots of pets out there to collect in zones around those levels. Would you ever consider leveling in this fashion?