After you do the introductory scenarios you get one piece (choice between head, chest or shoulders) that has an azerite talent tree. It has 3 talent rows that you gained at level 1, 4 and 7 of the Heart of Azeroth. (Row 1 of the talent tree gives you 3-5 choices (depending on class) - one that applies for all specs and 1 that is spec dependent, Row 2 gives you a choice between extra speed or a defensive talent and Row 3 gives you +5 ilvls).
After doing the quests for tirigarde sound and getting the lore achievement the final quest gives you a new piece of armor that has an azerite talent tree. (There are quests throughout the zone that have head, shoulder or chest armor reward but no attached talent tree - not sure if this will change). The choices for the new piece are different but the model is the same (ie 3 rows same sort of choices ). What is different is that you now get these talents unlock at levels 2, 5 and 8 or the Heart of Azeroth. I sort of feared this - as you unlocked better azerite armor you would need higher levels of the Heart of Azeroth to unlock the talents. That begins to feel like artifact level and that became a grind until changed later in Legion (which they said they had 'learned from'). I hope levelling Heart of Azeroth is not the same sort of grind.
The good is I now have 2 different procs (extra versatility and extra damage), and 2 different speed effects (one permanent, one on killing).