Classical guitar was played long before fingernails became the norm. However, for the tone and precision that an aspiring student of classical guitar seeks, fingernails are a MUST.

It is true that this will make tapping much more difficult. I have the same dilemma. I am following the Royal Conservatory of Music for classical guitar but at the same time love the rock stuff. It makes it harder but not impossible to tap. Unfortunately if you want to be a classical virtuoso, you will probably have to lay of the heavy tapping.

Fake fingernails... typically only used in case of an emergency (like a nail breaking before a gig or something). I had to put one on a few weeks ago when one of my nails broke before an exam. Also, don't fool yourself into thinking that fake nails are easy and you can pull 'em off and throw 'em on whenever you want to practice for half an hour. Shaping them is a BITCH! and they aren't to easy to get off... it's not a short term situation, but used for emergencys.

Like lalim said, if you are serious about classical guitar, grow your nails and get one of Aaron Shearer's classical methods for all the info on how to shape your nails for the maximum amount of volume and tone. If it's just a lark you're pursuing, or for occasional leisure playing, it might not be worth it.