Perhaps fifty percent of all raft trips are on the Trisuli, west of Kathmandu, and this is an obvious choice if you want whitewater with limited time or budget. Most itineraries are two or three days. The Trisuli has some rapids of medium difficulty (Class 3+) and good scenery, though it’s hardly wilderness – the main road to Kathmandu follows it the entire way, and in October and November you’ll have to share the river (and perhaps your beach campsite) with many other parties. Some operators have their own fixed campsites or lodges, ranging from private, green, semi-luxurious safari-style resorts to windblown village beaches complete with begging kids and scavenging dogs. Check out the camps and lodges carefully, especially with regard to how close a camp is to the noisy highway.
When booking, ask where the put-in point is: anything starting at Kuringhat or Mugling will, by and large, be a relaxing float. The best whitewater section is upstream of Mugling, from Charaundi to Kuringhat, and this can be done as a full-on half-day trip (perhaps as a break in the journey from Kathmandu to Pokhara).
The Trisuli lies between Pokhara, Kathmandu and Chitwan National Park, so it might make sense to incorporate your raft trip into your travel schedule. Your rafting company will normally be able to help you with logistics and look after your luggage. However, rafting all the way to Chitwan isn’t allowed, so you’ll have to travel from Narayanghat to the park by vehicle.