Kids of all ages are messy. The best thing I have found to help with the battle is a bulk package of white dishtowels, sometimes called bar towels. I usually get them at Costco in the tool/auto section.
Bar towels are bigger than a washcloth but not as big as a hand towel and since they are white you can bleach the heck out of them. I use them in the kitchen, to mop up spills, to do general cleaning, kid messes, dog messes, in the bathroom, just about everywhere.
The best thing is that you don't have to worry about staining a "good towel" when cleaning a bad mess. When they get too stained and ugly I use them outside and in the garage, washing the car, cleaning up paint spills, wiping down outdoor furniture, refinishing and staining furniture and other wood projects, the list goes on and on. They are definitely one thing I would not want to live without.
I would say the same for cotton diapers (the thick, prefolded kind) and a bulk pack of white terry washcloths. Cheap, bleachable, great for cleaning everything!