The Heart of Azeroth is an artifact given to the player character from the world soul via Magni Bronzebeard. It is a necklace that can use the power of Azerite, the lifeblood of the embryonic titan Azeroth (See Plot, below). Azerite is similar in function to Legion's artifact power, in that it is gathered normally from playing the game and is used to upgrade an artifact; likewise, the Heart is similar in function to Legion's artifact weapons by having the capacity to be constantly upgraded. Unlike artifact weapons, however, the Heart is shared with all of the player's specializations and can empower all eligible pieces worn simultaneously.
Congratulations on reaching Level 120! There are a myriad of new things to do. The first thing you should do is head to your faction hub ship where you have been doing your mission table quests and footholds to pick up and complete the quests necessary to unlock World Quests. World Quests will provide you an opportunity to grind reputation levels, acquire gear, and gather Azerite. There are tons of new things to do that we will cover in our Level 120 Battle Guide found here: Level 120 Battle Guide.
These guides are completely automated and work in-game, so you can see step-by-step instructions as you play. The guides automatically move to a new step as you complete the quests. You are told where to go i.e. the questgivers for accepting and turning in quests along with where and how to complete the quests. Zygor has more steps and details, which would be more suited to beginners. Dugi condenses the info into fewer steps, but still has all the necessary questing info.
Just like player item levels, the game's professions have become pretty bloated over time. Blizzard has made some updates to the leveling process before but Battle for Azeroth features a skill squish that should smooth things out considerably. Instead of having to level hundreds of points before you can start crafting Battle for Azeroth items, each expansion will have its own separate skill tier. In other words, you can start crafting Battle for Azeroth content right away, while the gear and items from other expansions will each feature their own separate counters. Your first Battle for Azeroth item will put you at 1/150 on the counter for the new expansion, no more grinding needed.
Meanwhile, Tyrande Whisperwind and Malfurion Stormrage grow impatient regarding liberating Darkshore from the Horde's control, ultimately choosing to begin the offensive on their own, against Anduin's advice. In order to maximize their chances, Tyrande undergoes a ritual to turn herself into the Night Warrior, the avatar of Elune's most wrathful and warlike aspects. To counter this, Sylvanas orders several Night Elven heroes who fell in the Burning of Teldrassil to be raised as Dark Rangers to bolster her forces.
For reference, my previous best time abusing Method-0 in 7.3.5 was 32.4 hours(as show in the old thread). A difference of just over 8 hours; which isn't all that bad, all things considered. And I believe I made several errors along the way, which could be cleaned up in future runs. I believe that that gap could be significantly closed by using a monk's bonus, but I hate playing monk...so...yeah. YMMV