Getting your class mount, in most cases, is primarily about just finishing the Broken Shore campaign, which is unfortunately gated behind champion missions, world quests, and rare spawns such that you might have to just log off and wait to complete them. The worst, however, is the hidden wyrmtongue cache quest. They did reduce this from having to find 10 to 5, but I really think it should be exactly one. Anyway, rant over.
Demon Hunters pretty quickly get a little scenario-like quest where they head to a remote fragment of Mardum and chase down the broodmother of the felbats on that world. You track it down and fight it while fending off its many offspring, but it mostly pretty simple.
Shamans simply get a summons from Thunderaan where you beat the three djinn bosses from Throne of the Four Winds (wait, if they're air elementals and you killed them in the raid, how the hell are they back? And if they can come back, why can't Al-Akir?) Anyway, these are just three tough but not too tough fights that I was able to do without much trouble. Once all three are defeated (not killed,) you get the mount.
And then Rogues.
Rogues have by far the toughest of the mount quests. I was lucky enough to encounter a couple of other rogues working on it, and teamwork definitely helped.
You receive word that the Legion has planted undead homunculi in all the major capital cities now that the Illidari are able to spot demonic infiltrators. To avoid putting the Uncrowned in a tough position, they send you to the opposite faction's cities so that your attacks can be passed off as standard Horde/Alliance aggression.
This is not phased or anything, so other players can make your life a living hell.
My rogue is Undead, so I went first to the Exodar, where you have to use Distract or Blind to keep the target's incredibly tough (basically he one-shots you) arcane golem from attacking.
Next is Darnassus, where the target is throwing a party in the entry area (from inner Teldrassil) and you have to sap her elite priest friend and then use Evasion and Cloak of Shadows while you target her and take her down (there's also a standard elite patroller who can detect through stealth that pats near the target.)
In Ironforge, there's a guy actually in the entryway to the throne room who takes almost no physical damage, so you'll definitely want to go either Assassination or Subtlety (and spec for Gloomblade for the latter) to max your non-physical damage.
Finally, the nastiest one that will never get easier: in Stormwind, you have to kill the guy in the Trade District auction house. So, where every Alliance player is in the city.
With those four killed, you're done, and you can be very, very proud of that mount.