Pretty much every hotel and decent-sized guest house in Chiang Mai can arrange for these trips, or you can stop into one of the ubiquitous shops offering trekking and other outdoor activities. Most of the outfits doing white water rafting day trips go to the Mae Tang. Chiang Mai Adventure also offers overnight trips that will take you to the Pai (200 km from Chiang Mai) or Wa (350 km away) rivers. Obviously the best rafting is during/just after the rain season.
Recommendations for elephant camps.