Cosby's Campground has an easy trail to swimming and a waterfall (Gabes Mountain Trail) The old settlers tr. is an extension of that and level. There is an old tree growth section near Maddron Bald. Huge old trees. Ttrails up to Mt. Le Conte are steep except for the Boulevard over from Ice Springs, but it is long. My kids at that age liked rthe camping much more than the hiking. Drive to Cades Cove if you can. See some old grist mills as part of a working museum. I think there are over 800 miles of trails in the Park!
Another up, but not very steep climb is the hike up to Mt. Cammere from Davenport Gap at the park boundary which is on a dirt road. The Davenport Gap shelter is on a mile in on the AT or you could start a mile furhte ut right at the Gap road. Spend the night at the shelter then day hike up to the tower.
Just north of Bryson City: A loop trail into a campsite 7.3 miles one way. Deep Creek, Indian Creek,and Martin's Gap Trails to Bryson Place Campsite #57. From that campsite next morning you head back on the Deep Creek trail 5.4 miles to the trailhead. Total mileage over moderate terrain, 12.7 miles. Encounter streams, coves, switchbacks and old road beds for nice hiking.