The first expansion quest kicks off in Silithus, so you can wait at Magni for release. He will give you the Heart of Azeroth and your first Azerite Armor Piece.
Level through your continent's 3 zones -
Suggested Alliance Order - Drustvar then Tiragarde Sound then Stormsong Valley
Suggested Horde Order - Zuldazar then Nazmir then Vol'dun
Save your foothold War Campaign quests for after you complete a zone. Doing all 3 gives about one level. After your first zone is complete return to your capital hub to start the quest.
Instead of vendoring quest rewards, take them to the scrapper after each zone. This will reward you with reagents and Expulsom (the new Blood of Sargeras).
At 120 you will mainly gear up by queuing for normal dungeons (item level 310 gear) until you have a item level 305 and can gear for heroic dungeons (item level 325 gear).
Make sure to do your 10 Regular Mythic Dungeons for the week. (Note that you can manually zone into heroic dungeons with a 5 man group under the required item level.)
You need to have all reputations to friendly to unlock Battle for Azeroth world quests. Finish up any lingering quests until you make this requirement.
Prioritize the Champions of Azeroth reputation as each rank gives you +15 item level to your Heart of Azeroth, as well as a 355 item level Azerite Armor piece at exalted. Purchase a Contract: Champions of Azeroth as soon as possible!
You will want to work on getting 7th Legion / Honorbound reputations to revered for the War Campaign Completion and to unlock the Allied Races.