My anchor is Wowpedia, starting with:
If you actually read down through that, it makes a structure that you can mentally hang the events on.
The Warcraft Universe is very very old, but all the events in the playable World of Warcraft have happened in the last 10 Azeroth-years or so, one Azeroth-year per expansion. It’s been a very busy decade on Azeroth! But there are lots o references to historical events, so having a hanfle on those helps you make sense of it all.
The lore in Shadowlands is unbelievably (and that’s a deliberate pun!) cack-handed and inconsistent, but the general structure, as laid out in the Chronicles, makes sense. WoW’s style over the years has been to carefully weave strands of the story little by little, where Shadowlands just dumped a heap of data from long long ago on us as if from a sack someone threw over their shoulder, all jumbled up with pieces broken, but the elements it dumped were not directly relevant to most of the story of World of Warcraft as shown in the game, so you don’t have to take the Shadowlands dreck in much detail.
So work your way down the Timeline, and click on the parts of the story you are interested in.
If you prefer a video approach, I recommend Nobbel’s channel on YouTube:
https://www.youtube.com/c/Nobbel87
Go to his Lore of Warcraft Playlists or his Quick Lore Fix sections and watch until your eyes are square, if video is your thing! And even if it isn’t, you can search his channel for a video about any particular incident or character that has caught your interest on Wowpedia.
Now: can you quest the full story? No. At least, it depends on what you mean by the full story. Because WoW covers only a decade of a millennia-long story, you obviously can’t experience, for example, the life of Arthas before he became the Lich King. (There is a critical moment in the Culling of Stratholme dungeon, but only a single event.) Similarly, you can’t experience the early story of Illidan Stormrage.
You can sort-of quest the full story from the start of Burning Crusade until Battle for Azeroth, and even Shadowlands if you want to do that, but you will want to also have a higher-level character to solo the dungeons and raids. So, for example, Wrath leads up to a confrontation with the Lich King, but you won’t be able to solo the three ICC dungeons that lead to Icecrown Citadel itself, or that raid. So the method would be to make a different character for each expansion. Play each through its expansion to level 50 (or higher) and then go back to complete the dungeons (some of which you might have done on the way) and the raids (which you definitely didn’t). A lot of the story is in the end-game instances, so you need those.
Also, some of the best work in the game, in characterisation and atmosphere, is in these instances, and they tie the story together and usually provide the final chapter, so you really don’t want to miss them.
This is a huge subject. If you have any more questions, please ask! I’m not a Loren-nerd myself, but I may be able to find answers.