Take a look at Xu-Fu’s power leveling guide and see if any of the methods fit for you. https://www.wow-petguide.com/index.php?m=Powerleveling
I’d say the absolute fastest way to level a battle pet to 25 is to use an Ultimate Battlestone. Requires 60 Polished Pet Charms each though. Might not be viable if you’re just starting out in Pet Battles and don’t have access to a large number of charms.
Other than that, battling repeatable trainers with a Safari Hat and pet treat buff. Even when the weekly even isn’t active, the trainers reward decent experience. Xu-Fu’s guide has a list of repeatable trainers.
My last pick would be battling wild pets. Find an area with high density and quick respawn time. I use my garrison a lot (the creepy crawly Halloween decorations), and Frostfire Ridge.