We used to walk up canyons, scrambling over rocks, lifting each other up and helping each other down and using an occasional stick, rope or boulder to traverse a more difficult bit. That's called canyoneering now and a full-scale sport most often specifically referring to that same scramble but with flowing water and, hopefully, a slot canyon involved.
Zion offers great canyoneering. The Narrows may be the birthplace of canyoneering. Climbing means ascending the face of a cliff using climbing gear and bouldering is scrambling up any rocky area sans rope.
Zion and the nearby area offer a wealth of climbing opportunities, both guided and unguided. Many rock climbing guides offer both group, private and custom trips, so you can tailor your adventure to your own level and that of your friends and family. Rock climbing is a hoot and quickly becomes addictive.
It seems there are two classes of climbers, those who prefer to challenge their ingenuity and their muscles (and their hand strength! Pack your squeeze ball in the car and do your reps while in the car, to get your hands in shape.) and those who love the thrill of going over a cliff. Rappelling is easy, once you get the hang of it.
Safety in the Canyons
The park service warns climbers in Zion to be wary. The sandstone cliffs can be unstable when wet and often look stronger than they are. The park doesn't permit any sport climbing, but the Visitors Center keeps a log of notes compiled by fellow climbers that can give you some insight into climbing opportunities within the park.
Zion requires permits for all adventures that use ropes as well as any that will include an overnight stay.
Be sure to check in with rangers before hiking The Narrows, as they may know about upcoming storms and other weather information that you should know about. Flash floods can come quickly (that's why they are called flash floods) and if you are in a narrow canyon there is little you can do to outrun it.
You also won't be able to stand your ground, so quickly climb as high as possible to get out of the way. Your only warning may be an increase of debris and a strange sound. If you observe either one, get to higher ground.