I have 2 sets of cycling clothes. I wash the clothes I've ridden in at the end of each day. I usually stay in a campground or motel and so I just take them into the shower with me and wash them with shampoo - simple, but I've also used sinks, standpipes and streams. The next day I wear my 2nd set of clothes and if anything isn't dry from the night before hang it off the bike to fully dry while I ride. Here are my padded under wear and socks arranged to dry while I ride.
Have you ever tried bringing a salad spinner? Just joking, of course, but tonight, as I was spinning some lettuce, I thought, "You know, this might work for some smart wool socks!"