I wash on warm days when they can dry in the sun though I do towel them off a little to get most of the wet off when their hair is long. If they've been shaved then I don't bother with the towel as they dry so fast anyway.
They wait until the warmest part of the day that they can wash them, let them shake, then walk them and tie them to the fence, use their hands under the belly going back and forth to help get some of the dripping water off, then towel dry. Stand behind the goat, and rub forward and backward on the back and sides then under the belly, working front to back. Works well for my kids.
They keep them tied in the sun so it helps them dry, once they are dry they turn them loose, or put them in a stall <or on the trailer if they are showing that day>.