Nobbel's channel is awesome, can't second that recommendation enough.
While there is an over-arching story, each zone or sometimes chains of zones have their own contained stories. Some of those stories are wholly experienced through seeing that zone's questing to completion, some of them require a dungeon to finish the narrative off. If you like the lore, the best way to absorb a lot of it is to simply play through all of the quests. You won't get all of the answers but you'll start to know what questions you want to ask.
One of my favourite experiences is the slow boil of leveling an orc through Durotar, the Northern Barrens, Ashenvale, the Stone Talon Mountains and The Southern Barrens. You start off this lowly peon killing boars, start working your way up in the Barrens to bigger things, then you join the fight against the Alliance in Ashenvale and become part of an orc warband in Stone Talon Mountains. You even get a rank and progress up in the warband. Then you see the war in the Southern Barrens, with the Alliance knocking on the gates of Mulgore.
I love that chain of progression. It's probably my favourite lower-level quest chain in the game, improved only by the Goblin starter area. Because old gods, sharks with lasers, towns in boxes, zombies and explosions.
The Forsaken questing experience is pretty great too. You become the right hand ghoul of Sylvannas WIndrunner, oversee the destruction of Gilneas and advance the cause of the Forsaken war machine through the Hillsbrad Foothills up to the Western Plaugelands.
WoD's story is pretty much like mists, you have to pick a character/area of interest and quest in it to get the story. If you want to learn what happens to Garrosh, Grommash, etc you have to play through the zones most closely connected to them and/or make significant progress through the legendary quest.
I'm playing through WotLK for the first time and what I find awesome is if you really see all of the questing through in a zone you get things like the Wrathgate cinematic or Arthas showing up everywhere. One in a dungeon most people just blow through in the dungeon finder and think nothing of. But if you quest through Grizzly Hills you basically discover that you're doing the work of the Lich King by cutting a bloody swathe through Drak'Tharon and his agent promises you immortality if you join him and Arthas.
Arthas is like this boogie man that you see from time to time while questing in Northrend. I don't think I saw Garrosh until that canned fight and cutscene in Nagrand.