Little or nothing right after a ride. A little chain lube on the chain and cleaner/lubricant spray elsewhere every so often. I'd like to do a better cleaning job once in a while, but I don't have anyplace to do it without getting soaking wet in sub-freezing weather.
Ironically, some of the cheap components on the bike are actually an advantage in the winter: The plastic gears in the Shimano Tourney derailleur don't rust or corrode.