Here's the general setup: One hand controls your mouse. If you have a relatively standard mouse, clicking the right button and whipping the mouse left to right will turn your character. Holding down both mouse buttons will cause your character to run forward.
Nearly everything else will be done with your other hand. There are a couple of exceptions that should be noted. First, for all that I keep saying "all your movement should be done with your mouse," I know many very successful PvP players who strafe on the keyboard. Second, some targeting duties will still fall back to your mouse -- things like targeting just the right enemy or hovering your mouse over a healing target.
Most of what your "keyboard hand" will be doing is executing damage rotations, kicking off defensive cooldowns, and triggering reactive abilities like interrupts.
The first step is to bind your old movement keys to be something else. Using a good, flashy effect that you will definitely notice is a good option; that will serve as a visual reminder when you've inadvertantly used the "wrong key."
Once you've done that, it's time to go for a jog. I'd recommend a leisurely stroll from the top of Kalimdor all the way down to Tanaris. Start your character running (on a ground mount's fine, but a flying mount rather defeats the point). By the time you've reached the bottom of Tanaris, you'll find turning left and right with your mouse has become a whole lot smoother.
I could tell you all of my keyboard buttons, but that won't help you; we use different spells. Instead, let's start focusing on the logic that drives how you should set up your keyboard.
As you're making your long jog along Kalimdor, pick a few fights along the way. What button do you use the most? Put that on your keyboard. Stop by a target dummy and practice your rotation. Everything you press should be somewhere on your keyboard.
I still use the Q and E keys for strafing, but my basic damage and tanking rotation goes on the old movement keys. (I back up using the tilde key [~].)
I put lesser-used spells where my fingers have to stretch further and less naturally hit the keys. Maybe I just have tiny-baby hands, but I tend not to easily hit the Y key with my keyboard hand. The letter G is about as far as I can make it.
Once I've got all the "big stuff" mapped out, I start setting modified keys. You can write macros and assign them to keyboard commands like Shift+2 or Ctrl+W.
Try it out in the battlegrounds
Before you start trying to mouse turn in raids or arenas, try it out in battlegrounds. The high demands on your reflexes and situational awareness should cement your habits into place.
The other thing BGs will do, though, is get you in the habit of turning quickly. With rogues and mages whipping around you at all times, you'll pick it up very quickly.
Healing with the mouse
Mouse turning is slightly more complicated for healers, especially if you make good use of raid frames. After all, you'll probably want to use your mouse to use most healing macros and special tricks.
Depending on how you have your UI set up, you can still turn with your mouse, and use the strafing keys to get you out of most trouble. Experiment with the setup and see how it works for you. Healing is, admittedly, a pretty big challenge compared to the other roles, at least in terms of moving with your mouse while actually casting your spells.
Keep at it
The last trick to learning how to mouse turn is to take your time. Don't get frustrated if it doesn't come naturally. I guess if it were easy, that'd be how the default control scheme would be set up. But it definitely has benefits, and once you've got the method learned, you'll be a lot better off.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.