If you want to use cotton, you should look for mercerised cotton. This has been treated to produce a non-fuzzing, slightly shiny appearance which is perfect for lace.
You might also look at blends with linen, hemp, nettle, or silk. There are also some remarkable yarns around which are cotton-like, but are made from bamboo, milk protein, crab shells … the world’s your oyster (and there’s also pearl yarn!).
Do not disdain wool: there are many wools around that are soft to the skin, and have wonderful stitch definition which is perfect for crochet. Specific breeds that are good are Blue-Faced Leicester and Falkland – I can particularly recommend Eden Cottage’s BFL Bowland and Debbie Bliss Falkland Aran – I designed this in Falkland Aran:
Incidentally, Ravelry is a great resource for learning about yarns. You can select to look at different weights, fibres, blends, sources, etc., and even look at projects worked in any yarn that interests you. Another great website is Yarnsub – especially if the price-tags on some yarns scare you! It’s still in its early stages, but still good. It was designed to help substitute yarns recommended in patterns if they were discontinued, hard to find, allergenic, or just to darn expensive!