Coolidge park is a good one, and is free. The Carousel is pretty cool with all the horses being made and maintained by locals.
If you are up for a drive, go up Signal Mountain and see the spaceship house, most folks forget that one.
Skip Ruby Falls, is a tourist trap (is a shaft drilled down trying to get at a lower cave, and most of the cave they opened has been damaged over they years due to tourists, and is not what it used to be, the lower cave they were attempting to drill to was used as a field hospital during the Civil War rumor has it that some famous folks defaced it with their signatures).
Point Park on top of Lookout is a nice place and has a small museum that you can hike down to. If you like hiking or driving around in the woods looking at scenery, Reflection Riding is a cool place, and they also have the red wolf program with their wildlife rehab program at the Nature Center (https://reflectionriding.org/). Over near the Nature Center is a plaque where one of the last battles of the Revolutionary war was at. You can Drive Missionary Ridge, lots of monuments up there, as well as some really nice houses, including one Frank Lloyd Wright designed, just be careful up there as ghetto is on one side. If you are here for the first Sunday of the month, the Hunter Art Museum is free then. They have some nice pieces and usually have some traveling exhibits as well.
The National Cemetery has a monument to the first medal of honor recipients there, which is why the museum is based here in Chattanooga.
I want to add, in my opinion, that Choo Choo BBQ (Hixson and East Ridge) is better than Sugar's although the fainting goats are cute, and I still prefer Momo's up in Dayton or Couch's over in Ooltewah. Bea's is an experience, if they are open while you are here, Southern cooking, served homestyle. As for food, will depend on your taste and what you like, there is a little of everything. The closest to authentic Mexican I have found, is at the hole in the wall behind the Mexican Grocery store on Main. Southern Star is pretty good, over near the Chattanoogan. Tony's up in the Art's district does homemade pasta, and is a good Italian place. Rembrandt's, again in the Bluffview Art district, is a great little coffee shop and has some yummy desserts.
This link has a calendar of what all is going on in Chattanooga by the week. https://www.visitchattanooga.com/events/