Tips for powerleveling here -
Battling gives ~2 times more exp than explore and takes a fraction of the time to complete, so its the best choice for powerleveling lots of pets. Pick out the pets you want to level to 50, and use all their energy on battling, as often as possible. Always pick the opponents, you need above a 50% win rate for battling to be more profitable than exploring, but try to stick to a 80% or higher win rate. Take note of who the gladiators are and fight them to keep from depleting the opponent energy stock. But don't pop the gladiators (aka make their energy hit 0) by losing or drawing with them too much, or you'll incur the the wrath of everyone after you who wants an opponent to battle.
Also, use autobattle to save time. Manual battle to boost win chance. And don't be too afraid of completely normal genetically powered pets - You can beat them if your pets have strong genes.
If you can't find anyone that you can win consistently against (50% or higher), just go back to explore and grab that bit of exp instead of letting the energy overflow and go to waste.
And if you have the money for it, the Riding hall is nice to get past levels where there's no opponents to battle. Its not worth it to use it for levels 1-10, but above that, the exp rate is great if you have the money for it.
Excluding the first couple levels, these are the averages I've experienced:
Riding hall: 1 level per 4 hours
Battling: 1 level per 10 energy
Explore: 1/4 level per 5 energy = 1/2 level per 10 energy
In terms of time spent on each pet:
Riding hall: A second
Battling: A few seconds
Explore: A couple minutes
Recent progress: Project Perfect
