We went to Zion with our family over Thanksgiving. Kids are ages 9, 7, and 6. There are several great hikes that even young kids can manage. We didn't attempt The Narrows although we did the easy hike that ends at the start of the Narrows. We did see kids doing it, but after talking to the park rangers, we decided it just would've been way too challenging for our kids. Plus, the rental gear is easily $50/person so $250+ for all 5 of us to just go in and then potentially have to turn back after a short time didn't seem worth it to me.
We stayed in Springdale at the Hampton Inn & Suites, which was great. Kids were in the heated outdoor pool every night, even in late November. Parking in Springdale is a little bit tricky but there is a shuttle that runs up and down the main road and we used that to get to the park and also to get to restaurants at night. There's another shuttle that takes you around the park.
We did find a cool slot canyon called Kanarra Creek Slot Canyon. It's outside of Zion, probably about 40 miles from Springdale. The water never got much higher than 6-8 inches deep, which might be different in the spring/summer. There are waterfalls and a ladder to climb. Water shoes or waterproof shoes are still a good idea because the water is really cold but it's gorgeous! It's also 3.5 miles round trip and at least half of that is the (dry and somewhat uninteresting) trail that takes you to where the slot canyon starts and then WOW! It was manageable for our kids and still a thrill for adults, too!
http://www.americansouthwest.net/slot_canyons/kanarra_creek/canyon.html