We just returned from our trip to Yellowstone! My husband booked us a 2 hour rafting trip for July 1st for a family of 4 (2 kids ages 8 and 11). Myself and our 8 year old were really nervous! But the staff go through everything and make you feel really comfortable. Our 8 year old chose not to sit on the edge of the raft and paddle and our guide, Keegan, calmly told her she could sit in the middle of the raft in front of me, which immediately put her at ease about the whole thing. Keegan was great and made us feel really safe. We hit class 3 rapids, which were not over the top for beginners or family with kids. Some parts of the 8 mile rafting experience were nice and calm where you can take in the scenery and ask questions about the area and the wildlife. When we needed to paddle, Keegan was clear with his directions. This ended up being one of our favorite experiences of our trip!
A couple of things to keep in mind -- The water temp of the river was cold (57 degrees on July 1st). They fit you with wet suits and shoes if you need them. Having been worn by others, the wet suits were a bit stinky (musty smelling), but they keep you warm so they do their job! If you have water shoes, bring them and wear your own! You will get splashed at times but we were in no way soaked. We went mid-day so the sun was up to keep us warm too. They don't allow you to take any belongings on the river for fear of losing it (including cell phones in a waterproof case around your neck). Hats and sunglasses were fine. So leave everything locked in your car. Since we did not have phones, we purchased the photo package for around $45 I think. We got 9 pictures that we were in, emailed to us. It was worth the money in my opinion, and were great photos of our family before heading down to the river, and while on the river in the rapids Great way to have some memories of our experience since we could not bring our own phones along and never would have been able to get the action shots of our raft on the river on our own anyway.