The Green River in Utah is one of the most popular destinations for whitewater rafting. It's a long canyon, and not all stretches are the same. If you want to introduce your family to whitewater rafting, one of the best stretches is the Desolation and Gray Canyon stretch, with over sixty Class 1 to 111 rapids. It's a 5 to 6 day trip, depending on how far you push each day.
Drive or fly to the Sand Wash ranger station for the put-in. If you want to raft on your own, you'll need to get permits from the various jurisdictions the Green passes through (mostly BLM, but also the National Park Service and others). If you bring your kids, it's easier to go with one of the many services that provide the permits, along with the raft, a guide, meals, and camp set-ups. It's also a great way to convince your wife or girlfriend that she'll enjoy the trip!
Once you leave the put-in, the water starts out smooth. As you proceed, you'll start hitting rapids, and it gets more and more interesting. The best spot for whitewater fans is the Belknap rapids, the result of a 1987 flash flood that dumped rocks from a side canyon into the water. If you're hardcore, ask the guide to bring an inflatable kayak so you can run that stretch on your own.
Bring a pack with a section to keep your wet clothes away from your dry clothes. Keep your feet happy with these Teva open sandals so they can stay cool in the warm weather. You can get matching Teva kids sandals if you bring the kids (check first; some guides accept kids as young as five; others take kids age eight and up), or give their feet more coverage with these, also from Teva, so they don't have to switch footgear when you take out to look at sites like an abandoned canyon, some amazing petroglyphs that are about 1000 years old, or the stone ruins of the long-gone Fremont cultures that made this canyon their home.