Step 1: Make a Slipknot
I plan to cast on 15 stitches, and I'm using pretty big (10 1/2 US) needles, so I'm going to leave two feet just to be safe. This is probably too much, but get used to overestimating, as it will save much time and frustration in the long run if you end up redoing it.
Make a loop around your hand and cross it so that you can see the end running to the ball of yarn through the loop. Pick up the "bar" inside the "circle" and tighten slightly by pulling the ball end of the yarn.
Place the loop on your needle so that the free end of yarn is closer to you.
You should be holding your needle like a bike handle bar. While maneuvering the needle may not be too comfortable at first, you should be fine just holding it this way.
Step 2: Cast on Your Second Stitch
Imagine the yarn hanging off the front and back of your needles are an archway. Put your left thumb and forefinger together and imagine they are a bug. Now, poke the bug's head through the archway.
Bad idea, the bug's head explodes and flies back at the arch! (separate your fingers, picking up the yarn on the backs of your fingers)
The magical arch seals beneath the remains of the bug's head. (draw the two ends to to your palm and hold them in place with your middle, ring, and pinky fingers).
ERrr.. so now your hand is in the right place. Check the picture.
Holding on to your slip knot with your right finger, lower the needle so that you can see a loop around your thumb and a loop around your foreringer. From left to right, there are four vertical sections of yarn comprising these loops. For the sake of clarity, I'll number them 1-4 from left to right.
Insert the tip of the needle under yarn 1, over yarn 2, under yarn 3. Using the tip of the needle, bring yarn 3 back between 1 and 2. Once this loop is on your needle, you can let go of the yarn in your left hand. Tighten this stitch gently so that it fits the needle.
Continue adding stitches in this way until you have the required number for you knitting.