Update: Some people are saying you don't need the starter quests, which is probably true -- there's a quest NPC inside the instance, so probably all you need to do is talk to him. But it is important to not have done the quest already that day -- pickup groups won't take you if you've already summoned him that day, as that means they get one less chance at the loot.
To start the quest, then, you just head into Blackrock Depths, and Blizzard has given us a nice mole machine shortcut just to the right of the entrance (I hear you have to kill some Dwarves there, but every time I've gone in, other party members have already cleared them out).
Etherjammer sent us some good tips he wrote up for the fight itself -- it's a pretty standard tank and spank, but there are lots of adds and other chaos to deal with, so it seems a lot more complicated than it is. EJ says you're supposed to clear out the Guzzler, but I didn't have to -- even with the occasional fear, we were able to keep Direbrew and his cronies in the room without grabbing more adds. There are three nonelites that come with him, and they pop up during the fight out of mole machines, tagging a group member for a few hundred damage each time. Direbrew also summons two Elite maidens -- one throws a beer at you that you have to drink to avoid a stun, and the other does an aggro wipe that's a little worse, but she drops fast if you focus on her. Both groups I've been in with just ignored both of the elites, but our healers were particularly well-geared so we didn't have any problems.
In general, drop the adds as best you can -- if you're fast enough on DPS and healing, you can let them build up a bit before clearing them out and then returning to the boss. And as EJ says, once you get Direbrew to 5%, you might as well pop trinkets and drop him, as everything despawns when he dies (the fight is generally long enough that you can hit trinkets at the start and hit them again right near the end). I was too chicken to try an all-out assault on him while ignoring the adds, but as a Hunter, I really only worried about them when they started hitting the healer, at which point I brought them down myself.
Here's one final tip, too: if you can, try to go every day with the same people -- almost all the stuff he drops is unique, so even if you lose a roll one day, if you take the same person the next day, the item is yours for sure. In two runs I haven't seen a mount yet, and the drop rate is supposedly pretty low, so if that's all you're farming for, good luck. But the other epic items seem just about Karazhan-level or above, so if you're in that early endgame zone that so many players (including myself) seem to be in, odds are you'll find something good that drops in there.