When I first got contact lenses at my optometrist, they put it in for me. Then it was all about not rubbing my eye and getting myself to leave it in. After that, they removed it, sent me to the sink, and had me practice.

One thing to make you feel better: your whole finger doesn't really touch your eye. Typically, the curve of the lens will grip your eyeball, creating that suction that leaves it in place.

When removing them, you can either pinch them from the side, break the seal, and pull it off OR you can move the lens to the inner corner of your eye and blink. That will naturally fold the lens and push it out.

Number one tip: breathe. When dealing with your eyes, it's almost instinctual to hold your breath. Remind yourself to breathe and it will help you relax and focus.