Hey Whisper,
I just joined the forum, but I live in Chattanooga. Moved here about 7 years ago from Middle TN. To answer your specific questions:
Chattanooga Gun Stores in my opinion are ok, but no great ones. I thought the stores in Mid TN were a lot better. There are several to choose from, but in my experience if you want something specific at prices that won't destroy your wallet, you gotta go box store or online. I've got a guy super close to me that does transfers pretty cheap, so a lot of times I go online.
Actual stores:
Carter Shooting Supplies in Harrison
Shooter's Supply in Hixson (FYI, one of their affiliated trainers spoke to local media against Constitutional Carry)
Southland Shooter's Supply in East Ridge
Shooter's Depot in Ringgold, GA (They had a TN store, but unfortunately it burned)
Cleveland Shooting Sports
Box stores: Bass Pro Shops in East Ridge, Sportsman's Warehouse (my personal favorite), and a couple of Academy's.
There are several places to shoot, but many of the best ones are about an hour or so from downtown.
Shooter's Supply gun store (25 yard indoor range, must shoot there ammo) Rifle and Pistol ok
Carter Shooting Supplies (25 yard indoor range, must shoot their targets) Rifle and Pistol ok
Shooter's Depot in Ringgold (don't know much about this one, I've never shot there)
Cleveland Shooting Sports (25 yard indoor range haven't shot here either, but a friend said it was a good range) Pistol only
TWRA Range in Prentice Cooper
Chattanooga Rifle Club (membership only, I've heard there is a waiting list, but I personally haven't check them out)
Montlake Sportsmen's Club Soddy Daisy (almost joined, looks like a well run club)
Dead Zero Shooting Park Spencer, TN (Hour fifteen from downtown, very nice ranges, out to 1k yards with electronic targets)
Mountain Top Gun and Archery, McMinville, TN (Hour fifteen from downtown). This is where I shoot. $10 day, no memberships. Pistol range and covered rifle range out to 500 yards. Most of their business if from hosting local trainers, so check their FB page to make sure the range isn't booked. When it is, you generally have it to yourself and you can get out on their rifle range and do some shooting and moving.
Gun Shows - very mediocre compared to what I've been to before
The Shriners host a gun show several times per year
Camp Jordan in East Ridge shows are probably once a quarter. I think they are RK affiliated.
Ft. Oglethorpe gun show (to be fair, I've never been to this one)
Welcome to the Scenic City! I'm always looking to meet up with fellow people of the gun, so if you have any other questions about the area, or want to grab a bite to eat some time, let me know.
Sempratus