On my SGIII, I'm currently using the CM10 Nightly (Cyanogenmod), however can be up and down with builds and changes daily. There are stable versions of CM that I would suggest checking out.
Another main ROM I have used is Omega, as it allows you to change up your kernel, take out all the bloatware on installation, and all through Omega Kitchen Files, easy to find anything you would like to Mod*at the simplest convenience.
Kernel would have to be Siyah stable and more battery life, but Stock is good enough on it's own.
Best place to check them out is Xda, can't go wrong; http://forum.xda-developers.com/forumdisplay.php?f=1563
Tbf, As long as you have ClockworkMod (CWM) and Titanium Backup, you're all set!
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