Breaching the Tomb

As a heads up to anyone who, like me, has multiple Alts. You DO need to complete this Achievement on every Alt in order to get the class hall mount.

Also, if you have not yet completed your class specific quest lines you will hit a dead spot about halfway through the Achievement where there will be no quests given out on Broken Shore. Complete your classes quest chain to the end (when you receive the achievement and Title for your class) and the next set of quests will open up and you shouldn't have any more dead zones.

(Edited to Update after SL based on below comment. I have not personally verified, Felt important to update)

From Andron827:
Since Shadowlands pre-patch you can use Chromie Time scaling to start Legion and your class hall way under level, but you will be gated at some point where the 'Glorious Campaign' or 'Breaching the Tomb' quests won't appear until you hit 45. This has happened so far on 3 toons for me, though I saw no mention of it elsewhere. If they fix/nerf this (or better yet make this achievement account-wide and skippable) I'll delete or update this comment.

*** INFORMATION FOR CURRENT (2020) PLAYERS LOOKING TO GET THEIR CLASS MOUNT QUICKLY***

TL;DR:
---Complete the Class Hall Campaign through "A Hero's Weapon".
---Complete the core Broken Shore quest line (not counting the dungeon/raid quests)

I just finished the class mount farm on 12 characters, all at max BFA level. Here's what you need to know to get it done without extraneous quests and garbage. (I had completed parts of the requirements on various alts, but several of them had done absolutely none of it. So it was confusing. If you did parts in the past, you can probably just pick up where you left off on Broken Isles and in your class hall.) Be sure to turn on "Trivial Quests" tracker so you can see all of the quests offered.

Also, I highly recommend doing this on all of your alts at the same time -- there are multiple places where you have to wait for a mission to complete or wait for a legion invasion to spawn, etc... when you reach a stopping point on one, move to the next alt...

Friendly reputation with all of the major Legion factions on at least one character of your account MAY be required for this. I'm not entirely sure. I was able to complete this on several characters with no Legion reputation whatsoever; my main has the rep.

• 1. Go to Dalaran (Broken Isles). If a quest pops up in your face automatically, or a NPC with a quest, accept that; it is your Class Hall starter quest. Get started on that long questline now or later... but you WILL have to complete that before you can finish THIS achievement and get your mount.
  
  
• 2. Enter The Violet Citadel and talk to Khadgar to complete Uniting the Isles. (IGNORE THE REST).
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• 3. Go to Krasus' Landing and Khadgar will be there. Continue doing his breadcrumb quests. He should offer you a chance to skip the scenario, so be sure to take him up on that.
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• 4. You end up at the quest area on Broken Shore. Get the two quests from COMMANDER CHAMBERS. None of the others are necessary, although I recommend accepting and immediately untracking all of them, so you can see when other new quests pop up.
  
  Anyway, your two quests: one is a travel quest. The other is to collect 100 resources, so complete 4 world quests along the way to get that done. (Rares and chests also drop the resources, but not very much. However, Rares DO drop the nethershards, which you'll need later, so kill them whenever you pass one)
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• 5. Turn in and do one more quest where you contribute to construction at the little table next to him. You can then talk to MAIEV SHADOWSONG to get Defending the Broken Isles. To complete this, you have to wait for a Legion Invasion to spawn in one of the four zones on the main continent (should be one every 18 hours, lasting for 6 hours each). Do one when it spawns.
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• 6. You cannot go further than this until you've completed your Class Hall Campaign through "A Hero's Weapon." Only then can you continue with the rest of this achievement. So don't get confused and think you have to complete those dungeon/raid quests or the daily quests at the very south tip of the island. Once you've finished with your class hall campaign, return here to complete all of the rest of the quests named in this achievement, one by one. (Ignore Khadgar's "Legion Threat" quests: unnecessary) Once you're done, you'll get the breadcrumb quest here at this quest hub to go for the class mount.
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  NOTE: Every class hall campaign is different, but all of them start with an quest or questgiving NPC that pops up when you enter Dalaran. You can follow this guide and click "view spoilers" for a list of the quests you have to finish, but you basically just do all* of the quests in your class hall...
  
  *There are a few exceptions: 1. You don't have to activate any artifacts weapons past the first one, 2. You don't have to do "Further Advancement" of your class hall after getting your title and doing A Hero's Weapon. 3. Ignore Khadgar if he shows up as a pink bubble with "Goddess Watch Over You". God, he's so damned annoying... really wanted to murder that pink bubble...
  
  Hunters can skip the Image of Mimiron quest to change Hati's appearance.
  Warlocks can skip "The Power Within."
  Rogues can skip the pickpocketting quests offered by Griftah and his genie.
  
  There may be more, but those are the ones I know about.
  
  P.S. If you simply don't have any quests you can find to further your class hall campaign, it's possible you dropped the quest chain at a point where you have to talk to some questgiver on the continent. This happened to one of my characters. I had completed the breadcrumb quest to go talk to some dude in Azsuna. I picked up his quests.... then I quit playing WoW for almost two years. When I returned, I cleared out my quest log and powerleveled up to BFA, not realizing I'd need that quest later. Go check out this guide, select your class, then click the link for "view spoilers" to see the long list of quests to find where you might have left off.

These are weekly series of quests on broken shore.

As personal opinion, most of these seem very lackluster and really makes me wonder why are they timegated.
List of parts that are just "do stuff you normally do":

Armies of Legionfall - Starting scenario
Begin Construction - talk to dude to help build a thing, available since week 1
Vengeance Point - talk to dude and kill few demons, available since week 1
Champions of Legionfall - Actual story for getting your new follower!
Mark of the Sentinax - Kill demons under sentinax for 100 "marks"..
Intolerable Infestation - "Kill 100 demons..again!"
Take Out the Head... - "Kill rare demons!"
Strike Them Down - "Kill 100 demons..err, again! yeah!"
Defending Broken Isles - Do invasion.
Shard Times - Give 2500 shards and get all but 1 back. Thats it.
Self-Fulfilling Prophecy - Do 12 WQs!
Relieved of Their Valuables - Probably somethign related to Wyrmtongue chests. I've heard it has story elements tho. Nope just chests.
Championing Our Cause - Rep related, most likely. Maybe get to exalted? Nah, just Order Hall stuff. Oopsie.