Take a gander at the sticky at the top of this forum... The guide to 5.3 tamer battles. Each team comp has room for a carried pet.
If I'm raising something from level 1, I do this (Alliance)
Evereesa in Swamp of Sorrows/random wild pet battle > Lydia Accoste at Karazhan > Stone Cold Trixxy in Winterspring > Morulu in Shattrath > Blood Knight Antari > Obalis > Aki the Chosen > whatever other tamer(s) I feel like until 25 (usually 1-2).
The whole route takes me about an hour (but I'm doing it on a mage with fast flying), and obviously can only be done once a day.
This is a modified strategy from http://fecklessleader.blogspot.com/p/1-25-veteran-pet-leveling-guide-10_27.html
. His guide is very good, but I'm trying to get more bags a day. He also has strategies for your first couple 25s, and an intermediate guide if you don't have access to the pets in the Veteran guide.
There are more daily tamers than I can reasonably do in my couple hours a day to play. Finding those teams that can carry a low level pet is key.