Some of my favorite day hikes are:
Porters Creek trail
Spruce Flats Falls trail
Andrews Bald trail

In general the Greenbrier Area (Porter's Creek) off 321 is a less crowded area. And there's some nice old growth trees there.

Spruce Flats Falls isn't on the park maps, but is one of the nicer falls in my opinion and an easy hike. Just ask the staff at the Smokey Mountain Institute at Tremont-the trail starts in that parking lot.

Andrews Bald is popular but is one of the higher elevation hikes that's also relatively easy. Nice view at the end.

If you're ready for a strenuous hike, consider the Siler's Bald trail. You'll be on the AT hiking the ridge with TN on one side of the trail and NC on the other side.

Cades Cove is great but perpetually crowded when school's out. A couple of days each week it's closed in the AM except for bikes, that may be a good solution.

Alum Cave Bluffs, Luarel Falls and the Chimney Tops are also very popular and very crowded as a rule. I haven't tried it yet, but if you have two cars, I've been told that the Road Prong trail is very nice and all downhill. Park one car at the Chimney Tops trail head and shuttle to the Road Prong trailhead at the top of the mountain. Hike downhill with a side trip to the Chimney Tops on the way down.

Pigeon Forge should always be avoided during the car show weekend-that's a nightmare. However there is plenty of shopping and restaurants there. Your kids will love it-plenty of tourist stuff, however tacky it might be. Gatlinburg is also always crowded but has a nice aquarium, I'm told.