Aside from blowing 60 Polished Pet Charms to buy the Ultimate Battle Training Stone that immediately levels a pet to 25, fastest method is probably fighting the daily pet trainers in Draenor and Pandaria, or trainer WQs in Legion. Especially during Pet Bonus Weeks when you can level a pet from 1 to 25 from just two trainers. Outside of Pet Week bonuses from wearing Safari Hat (10%), Lesser Pet Treat (25%), and Pet Treat (50%) speeds things up too. And they stack.
Downside is that some of these trainers are really difficult unless you have a good collection of 25’s already. With the right pets there are strategies from Wowhead and Xu-Fu’s (www.wow-petguide. com) that almost guarantee an easy win.
Looking at your pet collection, you may have to spend some time leveling the pets you have already the same way you’ve been doing, slowly.