I absolutely LOVE Chattanooga! I had been there prior to softball, but this will also be our third time at the Scenic City Tournament. The Summit of Softball Complex is beautiful.
I haven't been to the BBQ place that lewam3 mentioned above, but this is our favorite BBQ place -
http://www.hillbillywillys.com/ It's not by the Summit of Softball Complex, but not too far from Warner if you are playing there. It is a hole in the wall (in a plaza) that you go up and order from the register and then seat yourself, but we love it. If you have more than 2 people in your group, consider one of the family meals - lots of great southern food - including the sides.
Also, if you have some downtime, a couple of places to see on Lookout Mountain is Rock City and Ruby Falls. You can buy combo tickets to both places here -
https://tickets.lookoutmountain.com/Info.aspx?EventID=21 We've been to both places twice and love them both. A little pricey if you have more than a couple of people, but worth it if it's your first time going. Rock City is outdoors and beautiful with many sightseeing opportunities. Ruby Falls is an underground cave with a waterfall at the end - which is cool to go through, but also cool in temperature on the hot summer days of Tennessee - I think it is in the 50s or 60s underground, so take a jacket if you are prone to being cold.
My Mom has also done the incline, which you can also buy as a package ticket - I have not done it. She liked it a lot, but I didn't feel it would be worth the price. Still, where the incline is, you can get up on their decks for sightseeing and picture taking without having to buy a ticket to the actual incline. There are also other, free sightseeing stops on Lookout Mountain.
Final tip for Chattanooga - the only thing I'm not a fan of is the traffic on 75 can get quite busy and move slow. Probably no more than any other major city - I'm just not a city person. Just make sure if you are traveling to fields on 75, especially during rush hour times, then make sure you leave yourself extra time.
Good luck at the tournament and I hope you enjoy Chattanooga!