While many like to just completely relax while in Ubud, there are a number of "adventures" to sample if you're so inclined. My favorite activity by far is the white-water rafting down the Agung River just a short drive from town. For this, I highly recommend Sobek, although there are other operators.
Biking / Cycling Tours
Mountain biking and cycling seem to be the latest 'in' thing around Bali in general, and around Ubud in particular. I have done one bike trip out of Kuta, and it was quite fun, an easy ride mostly downhill through very pretty countryside. Sobek also does bike tours, or you can check out one of these other operators: