I’ve been over this a lot with new people coming to the game, or perhaps those who played the Warcraft games before WoW.
There isn’t a good way.
For a start, much of the Warcraft lore isn’t even in the game. It’s in the books, and/or the previous games, and/or the Chronicles, and/or Chris Metzen’s head, and/or whatever demented madness the writers at Blizzard Central are now cooking up.
The good news is that every zone has its own story. This is great. It allows players going through a zone to digest a manageable chunk of local lore and story as they level.
The problems come from
a) the end-game of each expansion
b) the stuff that isn’t in any expansion
The Operation Shieldwall and Dominance Offensive (Horde-side half of that story) are maybe the best example of end-game story. Do they still require the same amount of dailies to get each new story drop? Those stories work themselves out over the two years of an expansion’s max-level evolution, and give some sense of things moving on to people who are in the otherwise-limbo of being stuck at the same level.
There is also a nice nugget of end-game story in the three ICC dungeons before ICC itself, but it’s more condensed.
Some of the end-game stories have actually been removed. The whole Cloak line from Pandaria is gone now, as is the Ring line from Draenor.
Whatever, people levelling through the content will miss out on these main narratives of the expansion, because instead of doing the end-game content, and sometimes the pre-patch content, they will just immediately move on to the next expansion.
Some of the big stories literally can’t be seen on one character. For example, Operation Shieldwall is Alliance only, and Dominance Offensive is Horde only. Even though they cover the same events over the same time, the people playing through them see different things. And the split of Pandaren between the factions, and the joining with the rest of Azeroth, can only be seen on a Pandaren character.
Blizzard have been making end-game narrative more explicit since Mists. Here’s a question for you: what is the end-game story of Outland?
It’s more coherent than you might think, but it’s not explicitly in the game.
Operation Shieldwall/Dominance Offensive was patch 5.1, btw. After the two sides sent their elites (us) to Pandaria, it’s the story of the arrival of the main forces. Then, in the scenario Dagger in the Dark, which you can do solo at 120 if you ask the Lorewalker Scenario Storyteller above Mogushan, we find the point where Vol’jin, already about to break with Garrosh’s increasing craziness, gets pushed over the edge. Then in 5.2 we all have to prevent the disaster that the Zandalari have unleashed, and Jaina and Lor’themar establish an understaning that Garrosh Has To Go. 5.3 has been removed, but didn’t really have much story content anyhow; just bring Vol’jin supplies for the rebellion. And 5.4, of course, is the combined assault on Garrosh’s stronghold.
I’d say: look at where you are on that Timeline:
and read around Wowpedia and Wowhead to find the in-game quests that lead you through that content.