When you cross stitch on paper it looks like you cross stitched on paper. 8-) I don't like the look. Having said that, you can stitch any design on any fabric. Try stitching it on 14 or 16 count Aida cloth to start. Leave enough fabric so you can make a wallhanging, quilt, totebag, or stuff it and make a 3D snowman.
I used to cross stitch on my daughter's clothes. You use waste canvas to cross stitch the design and when finished, just pull out the strings of the waste canvas.
edit: check out "The Stitchery" for kits and "Herschnners" for fabrics. Great sites. If you really want to drool, check out "abcstitch". Take the virtual tour. And don't get mad at me after. :wink: