Azerite equipment is currently balanced around being able to unlock the outermost, earliest tier of power immediately upon acquisition, while delaying further advancement until the Heart has been further empowered. For example, the first piece of Azerite equipment, gained shortly after the Heart itself, is fully empowered once the Heart of Azeroth has reached level 3. A character at 120 (the maximum for the expansion) may find a piece whose first tier requires the Heart to have grown to level 12 or greater.
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.
Dugi’s leveling guide is the leveling addon I’ve always used to level as fast as possible. It will show you where to go and what to do in-game, speeding up the leveling process. This is optional though, as the game already does a pretty good job at showing you where to go at all times. The addon does, however, come with a couple of other nice features and perks.