The most hygienic restrooms are found in hotels and department stores, while those at most train or gas stations can be considerably less so. All are usually clearly marked with international symbols. You may encounter Japanese-style toilets, with bowls recessed into the floor, over which you squat facing the hood. This may take some getting used to, but it's completely sanitary as you don't come into direct contact with the facility. Train station restrooms often have at least one commode for the less agile at the end-row stall.
In many homes and Japanese-style public places, there will be a pair of slippers at the entrance to the restrooms. Change into these before entering the room, and change back when you exit.
Some public toilets don't have toilet paper, though there are dispensers where packets can be purchased for ¥50 (45¢) or so. Similarly, paper towel dispensers or hand dryers are not always installed, so a small handkerchief is useful to dry your hands.