I have this horrible preconception of certain things about raiding. I know it's wrong and I attempt to overcome it whenever possible.
Since the only time I ever really paid attention to Kiki's stats was when I was raiding TK and SSC... I've come to the notion that all raiders need those stats. That's what I did! That's what YOU should do too!
And I would be ... mistaken. (Not wrong, for I am never wrong.)
But one thing I am firmly against is 'gearing for Karazhan within Karazhan'. It caused problems, it will cause problems and it's generally a bad idea. (And it only works for DPS, unless your second rate healer/tank is partnered with a uber healer/tank that can pick up their slack, but generally, poor DPS is carry-able whereas a poor healer or tank is not.)
However, this doesn't mean that you have to run heroic after heroic after heroic after heroic after heroic to get all epics and badge gear before you ever step foot within Karazhan!
What this does mean is that you WILL have to run certain instances to get dropped gear, or you WILL have to craft gear, or you WILL have to get rep to get rep items, or you WILL have to PvP to get some needed hole in gear (damn you, druid tank shoulders! Damn you!) or you WILL have to buy gear. If you're an instance-hog, omg there are so many drops available to you to get you where you need to be. If you like heroics, please, do some. If you're a crafter, make whatever set is spec-appropriate for yourself. Etc. Etc. Ad Nauseum.
(If I could get Abs to show me how to do the wowhead links, like he promised, I've got some ideas for blogs I could write! But alas, alas, I cannot and he won't... *sob*)
Now, in order to be a successful run, REGARDLESS of the color of your gear, you need to have a certain amount of numbers. You can find a rogue in all purples that can't hit the broad side of a barn outside of PvP. This is because PvP attack gear is usually NOT what you want in PvE settings... and getting PvPers to understand this is sometimes related to banging your head against a wall repeatedly, while wishing it was THEIR head you were banging against a wall repeatedly. PvP attack gear usually doesn't have +hit or +spell hit on it, because in PvP, that isn't as important as attack power and crit rating and penetration and all that fun stuff I know next to nothing about. I don't care if you have 1 trillion attack power... it's useless if you can't hit your target.
Seriously, if you doubt the words that I say, get yourself a damage meter and play with it the next time you're in an instance or heroic or whatever. (not applicable against normal mobs, because normal mobs don't have the stats an instance or raid boss will have) And see. We've had someone in our group with amazing purple gear, he looked awesome. He had infinite cosmic attack power. And he was behind in DPS against two non-epixd players. Because his hit rating was through the floor.
Or you're a tank but you're stacking stamina instead of uncrittability, or ... I could go on.
But I digress!
(As you knew I would, so don't act all surprised.)
Here is a link to the recommended starting numbers for a Karazhan run. They aren't that hard to reach. And depending on how much leeway you're willing to give people, some of those numbers are fudgable, with buffs, kings, gotw, etc.
The other big thing that will help in your early runs is making sure everyone knows the pulls. You can find guides galore out there that discuss the strategy for the trash pulls and the bosses. There are probably as many different ways of doing those fights as there are ... uh, fingers on a hand. (I was going to say sand on a beach, but there probably aren't QUITE as many as that.) So just pick one that works for your particular group setup and run with it.
And above all.. have fun. It's a game.