Once you’ve stripped down, take a wash towel (provided) and wrap it around yourself. Japanese towels can be quite small sometimes, so if all else fails just hold it strategically in front of you where it will do the most good.
Then you can go into the main bathing room. Once in there, you will see a row of shower nozzles set low along the wall. There will also be some small plastic buckets and small plastic stools as well. Take a stool and position it in front of a shower nozzle, then sit down.
Grab a bucket or two. There should be soap and shampoo near the nozzle for your use. Sometimes the nozzle is just a faucet, which is where the plastic bucket comes in. Fill it up with water and dump it over yourself.
Once wet, clean yourself using the soap and shampoo provided. (If you prefer to bring your own soap and shampoo in, this is perfectly acceptable.) Then you can rinse your body and hair again by filling up the bucket as many times as necessary.
Use the wash towel to help scrub yourself if you like.
Once you’re clean, you can proceed to the next stage, which is the relaxing part. There will be at least one community bath, probably with other people already in it. (This may or may not be in another room.) Go ahead and get in, being mindful that the bath may be quite hot.
IT IS VITAL THAT YOU FULLY CLEAN YOURSELF BEFORE YOU GET INTO THIS BATH. The water is used by many people throughout the day, and this is how the Japanese keep it clean. Although the water is continuously being drained and replaced, no one wants to have it dirtied with soap and so on. So please use the community pool, whether onsen, rotenburo or sento, purely for relaxation, not for cleaning yourself.
Once you’ve enjoyed the bath as much as you like, get out and rinse yourself off again under a nozzle. Most likely you will be sweating a bit from the temperature. After you’ve rinsed off, take your wash towel and proceed back to the original changing room.
This room will very likely have hair driers, brushes, combs, and so on that, you can use. It will probably NOT have a regular towel for you to dry yourself with, so use the one you brought with you. Change back into your clothes (or yukata, if you’re staying at a Japanese-style hotel) and you’re done!
One thing to keep in mind is that there will likely be other Japanese people in the bath along with you, and they may be very frank about looking at you. It’s not considered particularly impolite in Japan to do this, so don’t let it bother you.
Some of them may also try to strike up conversations with you as well, which is also perfectly acceptable here.
The public bathing areas in Japan have traditionally doubled as public gathering places for centuries, and people make new friends in this manner all the time. Just go with the flow and you’ll be fine.