Hey Everyone,
If you cut them too short, it can be painful. I agree you don't want them digging into your fretboard, but you don't have to cut them to the quick. I usually leave between a 16th, and an 8th of an inch on mine. This helps with many of the techniques I use, doesn't seem like you're referring to any of these though. You will want to learn more eventually, and some of these other techniques will require a little bit of a nail.

The best thing to do is get an emery board, and slowly take them down to where they don't touch the fretboard. If you cut them too far back with nail clippers, there is no fixing them without a manicurist. Trust me, it's not easy to play with fake nails on your fretboard hand. Filing them down, makes it hard to get them too short, or cause pain. Nail clippers don't always cut evenly, and clipping in the wrong place is much easier to do with them. One little slip, and you can't play until the pain goes away, or the nail grows back out.

Have a good Memorial Day weekend, and best of luck in your learning.