Congratulations - I've used Looks in all their incarnations and have always found them cooler, classier, and miles ahead of Shimano. They do take a bit of getting used to compared to SPDs, but it is well worth it - the difference when I tried them on the winter singlespeed rather then its usual dual SPD 530 stunned.
Firstly, the weight distribution of the pedal means that its natural position is perpendicular to the road, with the binding face to the rear - where you want it.
If you miss then it does spin round, and you have to wait for it to settle down, but it does.
My usual reason for battering shins in the way you have is panic - and trying to engage too fast. Try touching the top of the pedal with the tip of your shoe. Press gently forward with the tip - parallel to the ground - until you feel the front edge of the cleat engage with the pedal. The gap between the cleat and the sole of your shoe will have caught on the raised bar at the front of the pedal. From there press gently down with your heel. You will have engaged. If that is taking too much effort, try loosening the tension to begin with.
If you skid off, wait about 10 seconds and have another go. If it still doesn't work - and there are those days - then you can pedal unengaged to get some speed up, and the confidence to try again without fearing that the motorist behind you isn't going to run you over.
It does become second nature after a couple of days.
Hope that helps.