I remember the first followup visit I had when I first got contacts and the doctor removing and placing them again on my eyes. And remarking to him how amazingly adept he was at it. I barely felt the lenses being touched. After a few months, I was able to do the same.
You should never actually touch your eye (or rarely, anyway). Just the tip of a finger to the lens should be enough to place it. Removing them, I use the tip of a finger to slide the lens down, then pinch the (soft) lens with my thumb.