Another way to make the A chord: use the index finger to make a "mini-barre" over the third and fourth strings (this requires that you bend the index finger back at the first knuckle); note the second string with the middle finger.

If you're playing a classical guitar with nylon strings, I don't see how you could be briusing your fingers, unless you're mashing down really, really hard. I've played cheap, steel-stringed guitars most of my life and have never had bruised fingertips. Try mashing down on the strings just enough next to the fret to get a clean tone from the string.