The most important thing to remember is that you do not need to learn to throw the frisbee very far. If you can throw a disk consistently for 10 feet or so, that is enough to make a positive contribution to the team.
Also remember that when you are learning to throw, you often get depressed about the fact that you have no aim. Aim is not important at this step. The important thing is to get the disk flying flat and spinning rapidly. Aim will come quickly once you've gotten this down.
Just as in tennis, there are two main throws: the forehand (aka flick, hence the name of our team) and the backhand. If you've thrown around with your friends, you have probably been throwing backhands.
For both these throws, the wrist action is the most important (far more important than using your arm). You will develop big wrist muscles playing ultimate, which can be beneficial in preventing or alleviating RSI problems. Fortunately these muscles aren't so noticeable, so as long as you don't run around too much you can remain as waif-like as you desire.
Stand facing the person to whom you are throwing. The foot opposite your throwing hand is called your pivot foot. You can step all you want with the other foot, but that pivot foot has to stay in place.
This is a top view:
Take a big step with your non-pivot foot. You should step forward and towards your pivot foot. If all goes well you now have your back somewhat turned towards your admiring potential throwee and are looking over your shoulder at them.
Here is where you step:
Snap your arm + wrist real fast (and let go of the frisbee). Ta-da! Your disk is flying gracefully through the air.
You are facing the person to whom you are throwing. For this throw, posture is important. Remember to have your shoulders over your hips in a pleasant rectangle. Your throwing elbow is glued to your rib (this is to keep you from using your arm to throw, a big no-no). You should be holding the frisbee as though it were a serving tray, with the side away from you pointing slightly towards the ground. This position will feel somewhat unnatural at first and is important to get used to.
Good flicking posture looks a little like this. The arrow shows the direction to step.
Take a little step out with your non-pivot foot (the one on the same side as the disk). Now snap your wrist and let go. It is important not to use your arm or you will never be able to aim; to compensate, really give your wrist some whiplash. It is strong; it can take it.
Do not panic if the disc falls pathetically to the ground. The important thing is to develop spin and to get the disc flat. The rest will come with time.