To learn about and experience Japanese papermaking, you can book at the Tenjin factory (Japanese). You'll have to visit it on a weekday. The charge is 500 Yen. If you don't speak Japanese, the best way to go about this is to work through the Uchiko Volunteer Guides for Foreigners. They can both make your booking and accompany you on your visit to tell you more about the history of papermaking in the area and the processes involved. It's also useful to have them as interpreters while you're making your sheet under the guidance of the craftswomen.
For more souvenirs than your own sheet of washi, pop across the street to the paper shop where you can find a wide variety of paper types and many beautifully printed papers—ideal for arts and crafts, home decoration, or as gifts.
If you have a keen interest in the paper trade, ask around or look at the websites of some local businesses and craftspeople that are trying to reinvigorate it. Products by Ikazaki Shachū include gilded wallpapers, and Wazm (Japanese) has designed some beautiful mobiles.