This advice was for an exchange student going on a brief homestay during summer.
I have very little information about using an onsen for guys, though it's fairly similar from what I hear.

1. When bathing, you bathe outside of the tub and soak in the tub, or ofuro, afterward.

2. Don't let soap or suds get into the ofuro water. Everyone in the house uses the same ofuro, so you have to keep the water clean. Sometimes this is hard when you have a shower next to the ofuro. If you do get any soap in the water, scoop it out quickly with a basin.

3. Don't let your hair get into the ofuro water.

4. Most private ofuro have some sort of shower in them, usually a hose with a shower head attached directly to the ofuro faucet. But there are many different designs of ofuro and faucets; if you are unsure what to do or how something works, ask. Usually your homestay will show you.

5. The typical bathing scenario is to sit on a low, plastic or wooden stool and dump basins of water over yourself to wash and rinse. With a large ofuro, like what you'd find at a hot spring or onsen, many people will use ofuro water to do this. But if you do that, you have to be careful to keep the basin from getting soap on it.

6. If you travel and do not stay in a Western-style hotel, you will likely end up in a public or semi-public bathing situation. If you are shy about this, cope. The vast majority of public ofuro are separate sex, so at least you won't have to worry about that.

7. If you are bathing in a public ofuro, do not stand up to shower.

8. Some people will stare at gaijin in an ofuro, but most women won't in my experience. I think, perhaps, if you're acclimated to Japan enough to use a public ofuro, then you aren't considered too foreign. Besides, staring is rude. I have never had a negative ofuro experience, and I've been all over Japan. Actually, I have many fond memories of “onsenning.” That said, my father—who is mistaken for Santa in the US and KFC’s Colonel in Japan—got plenty of stares when he went to the lovely onsens of Kinosaki.

9. In some public ofuro, people use small towels to cover their privates. I haven't actually seen this in person, but this is another watch and imitate situation.

10. If you go to a hotel with an onsen or hot spring, you will find various things in your room for you to use, including geta (wooden clogs), a yukata (light, summer kimono) and a tenzen (heavy, outer kimono). You are expected to wear these things to and from the onsen. Please remember when using yukata and kimono, the left side goes over, the right side under. The other way around is only for the dead. Among the things you'll find in your room, you should also have one of the little towels mentioned in 9. You can usually keep it. Most have the onsen's logo on it; I find they make great keepsakes.