My personal way of leveling pets is probably not terribly useful to a newer player.
How I used to do it though, was simply explore as much as possible and the rest of the energy battle. Nowadays I do it a bit differently:
Lower level pets I stick in the riding hall if I have the money (and usually only when I have a Trainer in my Inn, who halves the training fee), pets after lvl 30 go to explore once. If I don't lose any energy on the island, the pet goes to a second island. If I lose more than one fight and I end up with less than 5 energy, I battle the pet to other users' pets. Also if after all this the pet's exp is relatively far up and the training fee is less than 1000s, I train the pet. Mind you, I have level 5 Riding Hall, so I can train all the way to 50 if I'm so inclined.
Also, I interchange Druids, Trainers and Geneticists at my Inn, so I can capture better pets, research genes I want to know and train the pets cheaper.
I would be interested in what other players' strategies are? How do you prefer to level your pets and who do you invite to the Inn?