Well, rooting is just part one. Second stage would be a custom ROM.
But for now lets just focus on root. You can install Titanium backup, so you can remove any bloatware from your S3, backup data and apps and more.
You can use Greenify, Tasker, WifiKill, AdBlock plus for Android, Assistive zoom, Root Explorer, Juice Defender, Lucky Patcher, LMT Launcher, Network Log, FPS Meter (for FPS in games), DriveDroid, CPU Tuner, Good Mood Android Gesture Control, Undelete by Fahrbot, Call Master, SD Maid, ROM Manager, Samba File Sharing, Wireless ADB and more.
Custom Kernels also add a tone of functionality.
Just try the apps I've listed and see witch you find more useful for your needs.
Note that you will NOT find all these apps on the Google Play Store. Just google them and install the .apk
Good luck and enjoy your awesome rooted device!