When hovering over an ability, the tooltip will tell you whether or not that attack is weak or strong against certain pet types. Additionally, you can hover over the opposing team's pets and view their attacks during combat. This will be important since it’s a good idea to pit a pet that's strong against the opponent's pet who's vulnerable to that type.
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.