Blizzard will walk you through it
Hearing Blizzard talk about the Heroes of the Storm alpha yesterday renewed my interest in the game, leading me to this, a commentated match that serves as a nice introduction. Even as someone who doesn't play many MOBAs, I had little trouble following the action even during heated moments. That might not be a good thing depending on what you're hoping to get out of it.
The match itself happens on Dragon Shire, which revolves around two shrines that teams hope to hold in order to take control of the powerful Dragon Knight and push forward. I do quite like the look of Heroes of the Storm given my general inexperience with more demanding games in the genre. I mean, this looks like something I could, well, play.