The Taos Box, the 16-mile stretch of the Rio Grande south of John Dunn Bridge down to near Pilar [1], provides perhaps the best rafting in New Mexico, with Class III rapids ominously named Boat Reamer, Screaming Left-Hand Turn, and the like.
The river mellows out a bit south of the Taos Box and then leads into a shorter Class III section called the Racecourse—the most popular run, usually done as a half-day trip.
Beyond this, in the Orilla Verde Recreation Area around Pilar [1], the water is wide and flat, a place for a relaxing float with kids or other water newbies; you can flop in an inner tube if you really want to chill out.
Los Rios River Runners (575/776-8854, www.losriosriverrunners.com [2]) leads trips to all these spots.
Kokopelli, in particular, runs a daylong float trip in a lesser-traveled section of the river north of the Taos Box.
The best season is late May and early June, when the water is high from mountain runoff.
Links:
[1] http://www.moon.com/destinations/santa-fe-taos-albuquerque/santa-fe/outside-santa-fe/the-low-road-taos/pilar
[2] http://www.losriosriverrunners.com