Underwear: I find decent synthetic underwear more comfortable both on the bike and everywhere else, so that's all I have now. "ExOfficio Men's Give-n-Go Boxer Brief". My experience is that trying to bike with cotton underwear is sometimes a real bad time.

Cheap: Just any stretchy workout shorts/shirt.
Fashionable: Buy mountain bike shorts. Meant for biking, look like shorts, stylish. Expensive though.

If you feel that you need the padding that comes with shorts, mountain bike shorts have that as a removeable liner, and you can buy "underwear" style stuff that fits under any other shorts. I actually find riding more comfortable with a decent seat and no padding so I don't bother.