Lookout Mountain at the Georgia / Tennessee border can be a bit confusing to visit if you are not from the area. I stopped at the first place I came to driving up the mountain, Ruby Falls, and asked someone who worked there, "Where's the Lookout Point?" They did know of one public park that had a good view, but the best vistas are at the gated attraction called Rock City. It was just over 20 USD to get in, but there are services, family attractions, and plenty of well kept trails. Here is a link to the complete set of my Lookout Mountain Photos.
http://kimbriggs.com/photo/us-southeast/ga-lookout-mountain-fairyland
The panoramic photo at the top shows the best viewing area. The trails on the mountain top are a very unique place to walk in and around giant rock formations, bridges, and gardens. They are all easy to navigate.
The area was settled by some German immigrants in the early 20th century who had a love of German folklore. They added lots of gnomes until overcrowding caused a relocation to their own psychedelic section called Fairyland Caverns (I might be exaggerating a bit here), which is part of the Rock City experience.
If you are in the area, I think this is a very nice, family-friendly, vacation spot. Paying to get into Rock City is worth it for most people and you can get a discount if you visit both Ruby Falls and Rock City.
Resources
http://kimbriggs.com/photo/us-southeast/ga-lookout-mountain-rock-city
http://kimbriggs.com/photo/us-southeast/ga-lookout-mountain-fairyland