1. Good underwear. Look for those pricey high-tech long johns. Body Armor is a big brand name I think. Mosture wicking is the key term. Duck hunters, joggers, all kinds of sporting folks use these. Get socks and glove liners made out of same stuff
2. Wool socks and a wool sweater. Don't worry about staying dry while riding because it really isn't possible. Nothing is better wet than wool.
Because you're only riding 3 miles on a cheap bike, I suggest skipping any solution that costs more than $100. Although high tech gear from outfits like Cannondale or Pearl Izumi works really well for hardcore bike nuts, good wicking long johns ( $50), wool hunter's socks $6 (bright orange pulled over your pants cuffs) and a couple of Goodwill wool jumpers ($10) will work just fine.
Spend your money on better lighting for the bike if need be.