I don't think you really "become" an audiophile, as I believe the ability to judge fidelity is related to how your brain receives and organizes information and is something we're born with. It's certainly not related to hearing acuity and isn't something we can train to become if the basic framework isn't there.
But I know what you mean. I knew I was "special" when I could hear the difference between FM radio and the album on CD. I asked my father why the CD sounded better and he said he couldn't hear the difference. I think I was eight years old, so that was quite young. I became interested in audio around eleven, went through the requisite basshead stage for a while (with classical music, of all things), got my first "good" pair of headphones at 13, my first genuinely good set of speakers at 19, and my first (and still best) set of proper headphones at 22.
One thing hasn't changed. I still pay attention to bass. However, now what I look for are extension, texture, tightness, and balance; not hugely-enhanced and boomy car system bass. I actually prefer an analytical signature, but bass still stands out to me as one of the first things I evaluate, mainly because it's so hard to get it right.
Edited by Argyris - 12/22/12 at 12:54am