What type of cotton fabric to purchase?

I would look for something like twill or drill, poplin, butcher linen (which is "linen-look, not linen), pique or some other firm fabric for a tailored looking skirt like a straight skirt, an A line, a gored skirt, etc. If you want to do a skirt with gathers or ruffles, choose something softer like plisse, gauze, crinkle, lightweight muslin, lawn, seersucker.