It's not technically an attunement, but there is a series of quests that guide you through the introductory 5 Player Dungeons surrounding Icecrown Citadel. You don't have to do these dungeons to get into the normal versions of Icecrown, but it sure brings a lot to the experience. I guess, in a sense, you can think of it as a simplified attunement.
Additionally, the gear that drops from these first three instances will go a long way towards catching up people to the ICC raid. In my raid, we call this "bootstrapping." There's a bunch of reasons you might need to bootstrap folks: a tank rotation, new people, meteor crashing into the planet, what have you.
As you do the quest, you'll enter (and do battle in):
So, let's take a look at these three instances and how to blur through them.
(Note: I'll be mostly talking about the Horde version of the quest chain. However, there's an Alliance version as well, and I'll try and point it out along the way. Yes, I know - I'm horribly Horde biased, after an entire couple months being a Belf, compared to my four years of being a Human.)
You can pick up the breadcrumb quest in Dalaran, which instructs you to rush to Icecrown Citadel. This first quest is for The Forge of Souls, and sends you to meet Sylvanas or Jaina. You can either get there by manually hoofing it to the Citadel, or you can use the handy-dandy Dungeon Finder tools to teleport you there. I've not been in a group yet who doesn't use the teleport.
Once you've spoken to your faction-appropriate person, your first job will be to take down Bronjahm.
The Forge of Souls
The Forge of Souls looks like it's going to be a complicated maze, but most of that is just backdrop. If you stick to the path, it's pretty tough to get lost. You'll have a small variety of trash along the way, but nothing too troublesome. Keep to the main rule: Kill the one in the dress first. Most of the trash should just be handled by killing casters first, then melee mobs.
Bronjahm is the "Godfather of Souls." (His name is a mangled "James Brown" in reverse. Get it?) He's a caster-type boss with two distinct phases.
In the first phase, Bronjahm will create Corrupted Soul Fragments. They spawn on top of a random player. (What's happening here is that he's separating soul fragments from your character. As if you didn't have enough reasons to kill him.) Keep those soul fragments from reaching Bronjahm -- if they do, they'll heal him.
In the second phase, the boss will stat using Soulstorm. If you're on the edges of the platform, you'll take significant damage and be slowed. The way to avoid this effect is to move very close to the boss. (Hunters will need to keep a minimum range to attack, but they'll be safe at that minimum distance.)
Devourer of Souls
The Devourer of Souls is the second and last boss in the Forge. He has three tricks for which you should be prepared. First, he'll use Mirrored Soul. While linked with a player, damage taken by the Devourer will be shared to the targeted player. It's best to back off DPS during this time, and let the effect fade.
Second, the Devourer will Unleash Souls. A handful of ghosts will appear around the platform. While their damage isn't individually meaningful, if you have several of the critters pile up on you -- move. After this second phase, the Devourer will move on to Wailing Souls. It's basically an energy blast that sweeps around 90 degrees -- again, try not to stand in it.
Once you've downed the Devourer, you'll see Sylvanas (or Jaina) again, who will ask you to go into the Pit of Saron. There's a portal right behind where you killed the Devourer, and you can step through it into the Pit.