Iâve tried to think of the best answer for this question several times.
There is no good answer.
Maybe thatâs OK, because WoW is not a linear start-to-finish story. Itâs a mesh of different stories from different time, and just one part of a far longer universe (or universes.) 
The best answer I have, if you have a 120, is this:
Choosing: Make 1 Horde character and one Alliance character. One way to choose that might be interesting is to make one character of each race and level them to 5 or so, out of their starting zone. That doesnât take long, and lets you get a short feel of each character.
Classic: Quest through Kalimdor on your Horde character. Quest through Eastern Kingdoms on your Alliance character. You can do it the other way, but thatâs the most natural. Except, BE SURE to run your Horde character through Tirisfal Glades, regardless. It gives a key insight into the Forsaken. Oh, and level your Worgen until you get out of Gilneas. These two zones will explain the Greymane-Sylvanas relationship.
There isnât much end-game at 60, so you might as well proceed. By all means, see the Classic dungeons and raids, but I wouldnât say order was important.
60-80: Send one character to Outland and the other to Northrend. Quest through accordingly.
There is quite a bit that is interesting in Northrend at end game. You wonât be able to solo anything on your levelling character, but you can see it on your 120. Do the Heroic dungeons and raids in order - Wowpedia and Wowhead will give you the details. Especially, do the three ICC dungeons in order before ICC itself.
Outland isnât quite as interesting, and the story is not as coherent, but itâs still worth understanding why Magtheridon is below Hellfire Citadel, what Lady Vashj is up to, and to catch up with Gorefiend, whom you met in Shadowmoon, in Illidanâs Black Temple.
80-90: Same goes for Cataclysm and Mists. One character each. In Cataclysm, the raids just cap the stories which are almost entirely in the zones. In Mists, a lot of the story is told after you have finished questing. I note especially the great 5.1 storyline of Garrosh, Anduin, and the Bell. Isle of Thunder has an entire development and disposal of a huge character. Siege of Orgrimmar and the Scenarios is huge story-wise.
90-100: In Draenor, your Alliance and Horde will start in different zones, so that will get you half way through Draenor on each of them. You can safely split Gorgrond, Spires, Nagrand between them. There isnât much end-game story left in Draenor now, since Blizzard tore out the Legendary, but the opening of Tanaan Jungle is worth doing.