In addition to the aquarium, Chattanooga (& the surrounding area) has also a children's Creative Discovery Museum, the TN Railroad Museum, the Chickamauga Battlefield, Rock City, Ruby Falls, the River Park, a nice little zoo, & lots of different hiking & walking opportunities.
Chattanooga's restaurant scene has really improved over the past 5 years or so as well - lots of different dining opportunities.
In July or August, you'd be able to catch a Chattanooga Lookouts baseball game too.
The Lost Sea in Sweetwater, TN is about an hour or so away, & Fall Creek Falls State Park (free admission) is about 2 hours away.
A little warning, though - in July & August, Chattanooga, Memphis, & Nashville will most likely be hot & humid!