I've gotten two of the new class mount quests done. Both involve a short quest chain that can only be started after finishing the Broken Shore campaign (the last quest of which simply requires you to kill 100 demons in the zone - very easy if you go to the areas with lots of imps.)
The Paladin quest chain is highly reminiscent of the original Vanilla Charger quests (I never did the Blood Elf one added in BC, but I believe it had similar steps done in a different order.) Like that original chain, you are there to redeem a Death Knight's charger's spirit with the help of blessed barding. You need to pick up a few supplies (leystone and a Stonehide Barding from a leatherworker) and then you'll be able to go into Stratholme to face off against some necromancer who is doing some nefarious things.
The Death Knight chain is chapter two in the "uh... I'm not so sure we're the good guys anymore" series for the class. You get a message from the Lich King, who tells you of a vision he saw of a group of undead marching toward something on a glacier north of Icecrown. Yet when you go to investigate, there's only a tiny iceberg. The Lich King sends you to Wyrmrest to find out records of a fallen dragon in that area, and when the Red flight doesn't really cooperate, you go and take their records by force. You have an entire Ruby Sanctum's worth of red dragons and dragonkin to kill, and if you kill every last one of them (this might take a little patrolling to catch the flying drakes) you'll actually get a Feat of Strength achievement called Unholy Determination. Also, you'll be a pretty terrible person, as these guys' only reason to be hostile is that they don't want you raising one of their kind from the dead.
I'll be doing the Demon Hunter one relatively soon - I just need to get a Broken Shore champion mission completed - but the other classes will probably take a bit more time, as I haven't gotten them as far through the Broken Shore campaign (some haven't even gone there yet.)