I suspect your problem's down to the QR not being tight enough. If it's not done up tightly, the wheel can move under braking - it will indeed move to the left. The QR must be done up tight otherwise it's possible for the wheel to be ejected from the dropouts under heavy braking, at which point you'll go into the ground face first.
Yes, the right hand nut is the plastic one on the other side. Start with the QR lever in the fully open position, ideally pointing downwards. Tighten the nut up until you can move the QR lever about half way to the fork before you start feeling resistance. At that point push the lever fully in towards the fork. It'll need some force: a good guide is that, if you push it using the palm of your hand, it should leave a white mark on your skin afterwards.