Expect to find reasonably clean flushing toilets and cold running water at public restrooms in the major tourist destinations and attractions; toilet paper, soap, hot water, and paper towels aren't always available, though. Keep a packet of tissues with you at all times. At some tourist attractions, markets, bus stations, and the like, you usually have to pay 5 pesos to use the facilities. Since the H1N1 flu scare a few years ago, many restaurants, shops, and government offices have had hand sanitizer available for customers to use.
Remember that unless otherwise indicated you should put your used toilet paper in the wastebasket next to the toilet; many plumbing systems in Mexico still can't handle toilet paper.
Gas stations have public bathrooms—some tidy and others not so tidy. Alternatively, try popping into a restaurant, buying a little something (or not), and using the restroom, which will probably be simple but clean and adequately equipped.