I haven't taken my kids to the Smokies yet, but I have taken them (boys... 4 & 7) with me to Red River Gorge and I hike extensively in the Smokies, so maybe I can help on this one. If I were planning a trip with my boys, here is some advice that I would follow.
1. Get the most recent park map (available as a pdf on the NPS GSMNP website). Minor changes are made to the backcountry campsites and trails on almost a yearly basis.
2. Stay low. There are plenty of moderate trails down low that don't require the stiff climbs necessary to get up to the Appalachian Trail.
3. If you do want to see some higher elevation stuff, plan on starting at Newfound Gap or Clingman's Dome and hiking downhill. Arrange for a shuttle (available from local outfitters) so you don't have to climb back up.
4. I would try to avoid shelters and stick with campsites. You will have a little more privacy and a little more opportunity for the kids to explore and play.
5. Horse campsites and trails can be nasty. Lots of mud and horse poo. I just got back from a trip to the smokies where the only tent site left at a very crowded backcountry campsite was covered in horse poo, even though the horses were supposed to be picketed at a designated spot far away from the campsite. Had to invert a ziploc and clear the area. Still smelled like a barn.
6. I have found with my boys that 3 to 5 miles is all I should really plan to do per day. Yes, I can get them to do more, but they are really resistant to doing more the next day. I focus on what we can do for fun while we are walking and when we stop for breaks. Lots of suckers and snacks and maybe a surprise new toy hidden in the pack and brought out at the right time.
7. Tired kids always become excited and active the moment you stop for a break at a cool place like a stream crossing, a rocky area or an open view.
8. Lots of creek crossings in the smokies. Usually when you are close to a trailhead, there are footlogs and bridges to cross on, but the farther you get from trailheads, the more you have to do on your own. Bring a change of clothes for the kids and a pair of sport sandals that they can put on to cross the creeks. Great opportunity to stop and play for awhile.
9. If your kids haven't done this before, plan on an overnight trip or 2 nights at most. Also get a good recent trail guide book so you can learn about the trails you are going to hike before you go. This will help you learn about any other points of interest along the trail that the kids might enjoy (like the hidden cabin on jakes creek trail or one of the several wrecked trains or airplanes in the park).
10. If you have a gps, consider doing a treasure hunt for an easy geocache.
11. Reservations for campsites can be 1 month in advance of your trip. Shelters and some campsites require reservations to be made through the backcountry office. The majority of the campsites though only require you to fill out a permit, but no reservations are necessary. Regardless of if you need reservations, I would still call the backcountry office so the have a record of your trip. This way they can call you if any of your campsites later become closed due to bear activity or something like that (once they told me that I couldn't hike on one of the trails I was planning on hiking since it was closed due to a marijuana investigation.) You can try calling the backcountry reservation number (865-436-1231) but it is a farce... it is rarely active. Instead, call the backcountry information number (865-436-1297) to talk with a delightful volunteer who can help you plan your trip and make reservations.
Some ideas for a trip.
Start at Elkmont
day 1 - hike jakes creek trail to campsite 27.
day 2 - hike jakes creek trail, panther creek trail, middle prong trail to middle prong trailhead.
(note - requires a shuttle or 2 cars)
highlights - big cascade at the start of middle prong trail, lots of cool rock formations along middle prong, campsite 27 has cool boulders scattered throughout the campsite that the kids will love to play on, hidden cabin on jakes creek trail.
hope this helps.... glad to answer any other questions that come up.