Each pet has 3 abilities it can use during a battle, but a total of 6 to choose from. They will also have a total of 3 active spell slots that show which abilities can be used during a Pet Battle. The first spell slot will already be unlocked, but the second and third will only become available once the companion has reached a certain level. Other spells will also unlock after reaching higher levels.
2. Achieving. There are many pet battling achievements in game, and following the natural trail through the Taming [Continent] achievements up to Pandaria, and then on to the Spirit Tamers, Beasts of Fable and the Celestial Tournament was an epic journey for me and for many others. It's well worth doing. The Celestial Tournament is still, IMO, the pinnacle of PvE pet battling. However, WoD and Legion and BfA have added more battling achievements.