For those who have forgotten (i.e. your main did the intro questline ages ago and now you want an alt to have quick access to BfA for world quests or whatever):
1) accept this quest, head into Grommash Hold. A cinematic will start (which you can, of course skip).
2) head to the inn, wait for Nathanos to enter and then hand in the 'Battle for Azeroth: Mission Statement' quest to him.
3) Nathanos will then offer you 'The Stormwind Extraction' quest. Accept it (but don't drink the poison!)
4) speak to Nathanos again. He will now offer you the BfA intro skip.
5) take the skip and, after another skippable cinematic, you will find yourself in the Port of Zandalar next to Nathanos.
6) you can now hand in 'The Stormwind Extraction' quest to Nathanos.
Welcome to BfA!
For those that want maximum speed, there is another little skip available after this:
7) pick up 'Welcome to Zuldazar' from Princess Talanji (who will be just in front of you)
8) you have to slowly escort her and listen to her speak (you can't fly ahead)
9) hand in 'Welcome to Zuldazar' to General Jakra'zet.
10) accept the quest, 'Rastakhan' from General Jakra'zet BUT DO NOT MOUNT THE PTERRORDAX OR FLY YOURSELF TO KING RASTAKHAN!
11) instead, speak to Talanji and she will offer you another skip option. Take it.
12) another cinematic will start (which you can, of course, skip).
13) you will then find yourself next to King Rastakhan where you can hand him in the 'Rastakhan' quest and go on to establish your embassy. If you want, after speaking to the nearby NPC to start her walking and talking, you can fly down to the next level and wait for the Horde banner to appear instead of escorting her.
Welcome to BfA, Speaker of the Horde!