For whiterabbit25

Downtown there are the Creative Discovery Museum (which is having a Clifford the Big, Red Dog exhibit for the next few months), The Aquarium (the Aquarium has a man-made stream running through the outdoor part that my kids enjoy playing in), and the IMAX. On the North Shore are a couple of parks, both free,  that have beautiful scenery and fun things to do.  Coolidge and Renaissance are connected under the Market Street Bridge. Coolidge has a fountain that the kids can play in and also has a great carousel that is pretty cheap ($1/adults and 50 cents/kids over 3). Renaissance has a lot of pathways that the kids can run on and a couple of great hills to roll down. 

For getting around downtown, there is a free, electric shuttle with a map at every stop.  There is also the Walnut Street Bridge, which is a pedestrian bridge that is a nice way to walk to the North Shore. There is a water taxi, too, but I haven't been on it, yet.

Nightfall is a free concert series that starts every Friday night at 7:30. There is also the Chattanooga Market every Sunday, 12-5. The first Sunday of every month, admission at the Hunter Museum is free.

There are plenty of restaurants downtown and most people are very friendly, if you need directions or opinions on restaurants. 

If you go to Rock City, keep in mind that there are a lot of steep overlooks and drop-offs without decent/any railings. I keep a tight hold of my kids when we go there. Also, the main path isn't stroller friendly, although they do have an alternate pathway for strollers. It's a beautiful view, though. Point Park is at the top of the Incline and is a nice place to roam. Here's a good site for Lookout Mountain.

I can't think of a good place to hike, off the top of my head. Maybe some of my fellow residents can chime it?