I live very close to Pigeon Forge/Gatlinburg. Unlike other national parks, there is no charge to get into Great Smoky Mtn. national park. There are some great trails to hike. Laurel Falls trail is an easy one with a pretty waterfall at the end. Cades Cove is also very popular. Take a picnic pretty early to grab a picnic table by the creek. Take extra shoes & let your kids play in the creek, which is also a great way to cool off in the almost 100 degree weather we're having. After your picnic, drive around the Cades Cove loop, where you may see bears, deer, turkeys, etc... in an old Appalachian community.
Do a Google search for Ripley's Aquarium of the Smokies discounts before you go. I haven't heard one advertise on the radio this summer, but there are often discounts associated with a fast food restaurant, Food City, or Coke products. It really is a great aquarium; I prefer it to the Georgia Aquarium in Atlanta. Be sure to look for Lucy the sea turtle - she can be rather elusive!
The Apple Barn is a great place for lunch or dinner. Homemade apple fritters, apple julip, and apple butter are included with every meal. YUM!!
If your family likes baseball, our minor league team, the TN Smokies, have their stadium at exit 407 on the interstate (same exit for Sevierville, Pigeon Forge). This summer they have BOGO if you bring a Kraft singles package wrapper. I think they have fireworks just about every Friday too. They have a website as well to check out before you leave home.
There are A LOT of tourist trap dinner shows, museums, etc... in that area. You can blow as much money there as you would at Disney World! But you can spend hours in the park, just enjoying nature for free.