Other stretches are roadside and not as serene (although the massive amount of people sometimes make RH not that serene either). There are lots of options though. Lower Gunnison is comparable to RH and no permit. Less water though. I haven't done it but I bet because it is not permitted it can also get crowded. Back to the Colorado, all the stretches after Shoshone and before the diversion in Debeque can be floated. Not as nice as RH though
Moab Daily would probably be the next spot to take the kids. Lots of options for put in / take out depending on the length of trip you want.
You can go from Moab to Spanish Bottom and get taken out by a Jet Boat before hitting Cataract and that I is just a backcountry permit (which is probably free and not as restrictive). Best to use canoes though or have a motor on your raft.
This website is a GREAT resource for Western river floats. http://org.coloradomesa.edu/~jerry/guide/