Mostly, I get by fine with sneakers and platform pedals. I also use Superfeet insoles (now) or custom orthotics (when I learned to ride a bike). Those help make a regular sneaker much stiffer in the sole. My feet will screech at me if I start doing something silly like pedaling heels up, or with my feet in the wrong spot on the pedal. Fix up my form, pain goes away.

Most people don't have such whiny feet. So getting good form can be harder. Clipless pedals, toeclips and so forth are to help you get good form.