Well, anything you'd wear to an SCA event. In general, use woven natural fabrics, such as cotton, linen or wool. Veils should be silk or very fine linen, but modern imitations will also look ok. If you're not sure what sort of hat you want, or don't trust your design skills, copy one of the ones in this handout, or from a medieval illustration. Cut a piece of fabric 1 yard long by whatever diameter roll you want.
Some folks swear by cutting on the bias and stiff stuffing to eliminate interior wrikles.
There aren't a lot of wrinkles with this on the grain method. Stuff with stuffing. Try a tall thin drinking glass or other cylindrical object. The idea with the stuffing is to get it firmly enough to keep the shape, but not so dense that it can't compress very well. You'll have to experiment.