It's intuitive... to a certain extent... but one should use a light touch when applying makeup or beauty products in general. But let's further explore this.
The eye area is extremely sensitive and there are lots of tiny blood vessels and all and, well, your eyes themselves under the eyelids, and that all can easily get irritated or damaged. So, definitely, every time you apply something around they eye area or on the eyelid, use a very light touch.
It's a little known fact but your ring fingers (on either hand) are your weakest fingers. (I know this from experience both with makeup and with piano lessons... even through practice of using the ring finger, it's still weaker than the other ones.) So you will want to use the ring fingers to apply products around or on the eyelids.
Eye creams for like wrinkle prevention or treatment should indeed be applied with your ring finger. And did you know that you're not actually supposed to apply them on your eyelids? Eye creams should be applied under the lower eyelid (slightly above the cheek bone) and on the brow bone. Your eyes blinking will make the cream go up on the eyelid all on its own. It's less damaging for your eyelids to not apply creams directly on them. And you run less of a chance to accidentally have some product go into your eyes.
Pretty much whenever you'll apply something to the eye area, you will want to use little dabs motions instead of back and forth motions. So, for example, for eye cream, you will put a little bit of product on your ring finger and pat gently to apply. When applying eye primer, same thing but on the eyelid (this time it's okay to apply directly on the eyelid... it's the point ;)).
The same applies too for eye shadow application on the eyelid: you want to use little dabs. The back and forth motion will yield lots of fall out and will actually create a less even application (and won't deposit all the product that you've picked up, obviously, since a portion is flying in the air). This past weekend, I showed a friend the difference between the two. I should have taken a picture.
This may be counterintuitive but a gentle touch will yield much better results when cleaning up/removing your makeup than a more forceful stroke. So, if you find a stubborn spot (happens all the time), instead of rubbing harder and faster, go slower and more gently. You'll be surprised with the results and how it will come out more easily. The thing is that, if you are going hard at the spot, instead of removing the makeup (which is really trying to "lift" the makeup off your face), you're actually embedding it more in the pores... hence why it's harder to make it come off. And it's the same thing when you're trying to remove nail polish, by the way: when hitting a stubborn spot, go gently instead of harder.