The number one reason for hangnails is that your cuticles are too dry. When you see ragged cuticles or hangnails, it's your body's way of telling you to apply hand cream or lotion. No one moisturizes their cuticles enough. I tell people to put a bunch of moisturizers with pumps around the house, and whenever you come across one, use it.
I'm of the camp that you don't cut your cuticles. The cuticle is a barrier to keep bacteria from entering your body, so it's best to keep it where it is. On the other hand, you must groom them -- but that's easy. All you need is a good cuticle remover and an orange stick. Don't use a cuticle oil or cream for this -- you really need a cuticle remover to sufficiently soften the cuticle. After you apply it, gently push back cuticles with the orange stick. Then move the orange stick in tiny circles around the base of the nail to help remove the dead skin and debris that may be there. Do this at least three times per nail. When the debris is lifted, wipe it with a cotton ball. Then moisturize.