I remember when we were gearing up for Karazhan and trying to gather all the information possible. Is our gear good enough? Do we have the right classes? How far in should we expect to go? So, I thought I'd throw together a few observations on those questions you may have if you're first venturing into Karazhan.
We generally go with the philosophy of wanting one from each class, and then one additional member. Keep in mind, this is an "ideal" list, and rarely happens in practice. Things to keep in mind: You'll need 2 tanks, 3 healers and 5 DPS. You can get away with 2 healers for some of the easier battles, but to make things easy, this is what you'll need.
You may also have some of your hybrids play a duel role at some point. We generally have our pally tank some of the undead groups in the Ballroom right before Moroes. He can usually hold the large, undead groups better than our other tanks, so it speeds up the AoE process with less casualties. Also, a druid for a second tank usually has the gear to go full dps on the fights such as Shade of Aran or Prince where only 1 tank is needed.
You can certainly mix up the groups with different classes. Two or three priests for shackling in Moroes' room certainly doesn't hurt... being able to banish two or more of the elementals in the Shade fight... these aren't bad things. You mainly need to make sure that you stick with the basic rule of 2/3/5. If you have 4 prot tanks, you may want to think about letting 2 of them sit out.
What is needed? This is kind of a hard question to answer. The response question may be: how far do you want to go? You can probably do the animal bosses the first time in to Karazhan. To get attuned to Kara, you had to do enough in instances that you're probably geared enough for at least those. They may even be your groups first purple boots, bracers, and/or belt. The next few fights mainly take enough coordination, with Moroes and the Opera Event ramping up even more. By the time you're hitting Moroes, you probably shouldn't have any greens except maybe a trinket or two. And in the nights you're not running Karazhan, you should be getting the rest of your Heroic Keys and running Heroic instances. You'll should probably get a feel for what each boss will need as you attempt them. Give them a few attempts each, and don't be afraid to put them off to next week because you need more drops from Maiden or something like that.
This one may depend on how much money you want to spend. Flasks are going to be great to use while you're learning these bosses. They persist through death and last for 2 hours. You'll also want an ample supply of Health/Mana pots, food (stam, +dmg or +healing), and mana/wizard oil. As you gear out, you'll probably need less of these things, but it's better to be over prepared and one-shot 3 bosses then having to keep corpse running back.
Those were my initial thoughts of beginning to do Karazhan. Anyone else have ideas for the basics?