Welcome. This is my first post, too.
What Frisco said should greatly improve your performance. I have an S3 with ATT as well, and had the same experience as you (overall lag, typing lag, etc.). I wiped the cache and immediately saw a jump in performance. You may also want to try a battery pull: remove the battery while the phone is on. It's somewhat like a soft reset.
Despite all of the little "tricks," though, I wasn't happy with the performance. I decided to root the phone and remove some bloatware/ATT apps. The phone was still slower than usual, which caused me to pull the trigger on installing a custom rom.
I'm currently running the stable version of CyanogenMod (CM 10.1), and I'm extremely pleased with the boost in performance and overall "cleanliness" compared to the cluttered TouchWiz look.
If you're happy with TouchWiz, I'd say stick to wiping the cache and battery pulls every few days. If you're thinking about jumping ship and trying a custom rom, give CM10 a shot. The rooting process is relatively easy and can be reversed.
Best of luck!