Baths are not for cleaning in, as additional bathers split the water. You ought to clean manually before incoming the bath and there is an recognized routine for doing this whether an onsen, rotemburo or an ofuro (a bath in someone’s home). 
Firstly, I recommend learning the Japanese characters for men (otoko) and women (onna). Not all onsen have English written on the door, and you could save yourself a few blushes. Also bear in mind that some hotels only have one outdoor bath and so this is rotated daily. Do check that the ladies bath of the night before hasn’t changed to the gentlemen’s in the morning. Bathing is rarely mixed, and when it is, the bathers are usually not a day under 80. Separated bathing is the standard case, although many hotels offer a “family room”.
Once the correct changing room has been established, you should arm yourself with a small towel. This is either provided in the changing room or will have been offered at reception. After spending time in Japan you will find your collection of small towels starts to grow. You MUST remove your shoes at the door and leave them on the shelves or in the lockers provided.