Most of the water that gets your feet wet comes from the front tire. A full coverage fender does better than any shoe cover keeping your feet dry, except in downpours. Shoe covers will stop some of that but I've found that water will eventually find its way in. Like others have said, get shoes and socks that dry quickly and always try to have a pair of dry socks to slip into at the end of the day.
I like lightweight running shoes, Altras for me, and large flat pedals with pins (Oneup composites are grippy and cheap).