Lots of people that camp love staying at Fall Creek Falls 35-40 minutes away but that is more a a destination on it's own outside Chattanooga toward Nashville.
https://tnstateparks.com/parks/fall-creek-falls
It's great to go before it gets too hot but you can swim the falls, fish the lake, hike, golf, restaurant on site, and the falls are great. It's close enough you could drive into the city to do a couple things.
I love Chattanooga myself as my wife was from there. Lots of stuff to do and I agree the Aquarium is on that list of must do's. Driving up lookout mountain for the views (you see several states up there) was a do for me but I also liked Ruby Falls and Rock City. Ruby Falls is simple but I love caves so I may have enjoyed it more than most.