Remember: Level doesn't count in the Battle Dome. You want to raise your pet's stats as far as they can go before you need to raise the level. I see a lot of new people ask how to get level ups and using other pet's levels to determine if they should battle or not. Hit Points are what matter the most, in addition to the Training School you can feed your pet a Faerie Queen Negg from the Neggery and they will gain 1 Hit Point.
W: Level does count when you are earning Faerie Abilities.

Tip: Raise your pets HP as high as it can go. Then work on Defence then Power. If you start working on Defence and your pet gets a Level Up go back and raise it's HP more. Dubloons cost less than half of what Codestones do so I would recommend using Swashbuckling Academy on Krawk Island. But If you're in a hurry to raise them and NP doesn't matter go for the Mystic Training School on Mystery Island. Don't worry about raising speed it does nothing for your pet.