(I wrote this a while ago for r/wow, I thought I'd post it here as well)
I've seen this question being asked often since DF release so I thought I'd group all the options in a single post.

1. enable tracking of trivial quests
2. do not pick up more than one starting quest at the same time, you may be phased out from NPCs and unable to progress, if this happens, abandon all starting quests and pick again the one you want to do
3. if you intend to do all quests in old zones, buy a foam sword, it won't kill trivial enemies

Shadowlands

You don't get the quest anymore by zoning in your capital city.

Alliance
Go to Stormwind, on the stairs to Stormwind Keep there's a group of death knights with Darion Mograine among them, he will give you the starting quest Shadowlands: A Chilling Summons.

Horde
Go to Orgrimmar, in front of Grommash Hold (Valley of Strenght) there's a group of death knights with Darion Mograine among them, he will give you the starting quest Shadowlands: A Chilling Summons.

BFA

Alliance/Horde
The starting quest is given by an NPC standing next to a Hero/Command Board in Stormwind/Orgrimmar (not the board itself).

The quest is Tides of War for Alliance and Mission Statement for Horde.

Legion

Alliance/Horde
Go to Stormwind/Orgrimmar, click on a Hero/Command Board and choose Broken Shore/Fighting the Legion.

If the option is not there, pick up all irrelevant quests until "Broken Shore/Fighting the Legion" shows up, pick that and abandon the rest.

WoD

Alliance/Horde
Go to Stormwind/Orgrimmar, click on a Hero/Command Board and choose Tanaan Jungle/Enter the Dark Portal.

If the option is not there, travel to the Dark Portal in Blasted Lands, talk to Kadghar and start the quest "The Dark Portal".

Fast travel to Blasted Lands
Talk to "Vanguard Battlemage". For Alliance she's at the entrance of the Mage Tower in Stormwind, for Horde she's on the bottom floor of the portal room in Orgrimmar next to the portal to Warspear, the option is: I must report to the Dark Portal, it will teleport you directly next to Kadghar (you need the quest from the Hero/Command board for this option to show up).

If for some reason the area is phased to TBC, you'll end up in Outland, you can't go back through the Dark Portal but there are portals in front of you to go back to Stormwind/Orgrimmar.

Travel to Blasted Lands by other means (see bullets below) then look for a chat bubble icon on the upper part of the map, that's Zidormi, go talk to her and she'll make you change phase.
Alternatevely you can go back to a Hero/Command board, pick up all irrelevant quests until "Tanaan Jungle/Enter the Dark Portal" shows up, pick that and abandon the rest.

• Stormwind: take the flightmaster to The Harborage (Swamp of Sorrows) then fly south to Zidormi.
• Orgrimmar: take the zeppeling to Grom'gol (Stranglethorn) then either fly yourself to Blasted Lands or take the flightmaster to Stonard (Swamp of Sorrows) then fly to Zidormi.

MoP

Alliance
Go to Stormwind Keep, next to the throne talk to Grand Admiral Jes-Tereth and pick up the quest The King's Command.

If she doesn't have the quest, go to a Hero Board and pick up all irrelevant quests until "The Jade Forest/Report to Stormwind Keep" shows up, pick that and abandon the rest.

Horde
Go inside Grommash Hold in Orgrimmar (Valley of Strenght), talk to General Nazgrim and pick up the quest The Art of War.

If he doesn't offer you the quest, go to a Hero Board and pick up all irrelevant quests until "The Jade Forest/Report to Grommash Hold" shows up, pick that and abandon the rest.

Previous xpacs

There are no intro scenarios, just travel there and start questing.

Note for Outland: Blasted Lands is usually phased to the WoD version, in that case go talk to Zidormi (chat bubble icon on the map) and choose "show me blasted lands before the invasion" then you can enter the Dark Portal to go to Outland.
To fast travel there you can talk to "Honor Hold Mage" at the entrance of the Stormwind Mage Tower for Alliance, and "Thrallmar Mage" on the lower floor of the Orgrimmar portal room next to the portal to Shattrath for Horde.