My first suggestion to you is, when you're playing WoW, start clicking on books. There are books all over the game world, in various dungeons, out in the open, hidden away in corners in inns and cities. If you see one, click on it. Hidden in all these books are stories about the world of Azeroth. You can get surprisingly caught up on the basics just via this exercise. Also try to click on plaques on statues and in structures. The statues in front of Stormwind have small plaques on them that tell you exactly who these are statues of and why they were placed there.
There's also the expanded universe section of the official World of Warcraft site to consider. Not only is it home to the leader short stories Blizzard has been producing, but it contains a history of the Warcraft setting and links to the Warcraft tie-in comics, manga and novels as well for ease of purchase. It even preserves short stories previously posted on the site. It's a solidly organized central location for you to branch out.
While you're at it, you should also consider playing the previous Warcraft games. At the very least, you can still purchase Warcraft III at the Blizzard site. It still holds up very well for a game that's more than a decade old, and its story cutscenes explain the world just before and during the Third War. Blizzard does a lot toward making its game settings approachable, and you can definitely make good use of the resources on that site.
Hopefully, this will get you started on discovering the story behind why Deathwing hates Stormwind so much, who that Arthas guy was and why everyone was so upset with him, or why Alexstrasza dresses like that. If you figure out that last part, let me know.
World of Warcraft: Cataclysm has destroyed Azeroth as we know it; nothing is the same! In WoW Insider's Guide to Cataclysm, you can find out everything you need to know about WoW's third expansion, from leveling up a new goblin or worgen to breaking news and strategies on endgame play.