Congrats!
Try this ROM, its an official PAC-ROM (the best of AOKP, ParanoidAndroid, CyanogenMod).
PAC in Black, is particularly nice.

You can even play with the phone's kernel, r00t access, its governor, its voltages (decrease it), its clockspeed (increase it), gpu clockspeed and voltage, user interface and so much more.

Do things that are on system first to get better stability, response and battery life out of them. But for additional features that such custom roms might leave out can be alleviated with... The Xposed Module!

There's heaps of little nifty stuff (my usual are XPrivacy, Adblock, Tinted Status Bar, XBlast Tools, GravityBox, Xposed Additions).

Try to go for a clean look and functionality, overboarding on features really has its toll on the user interface AND battery life.

SGS3 is a pretty modern phone... its not as good as the Nexus 5, but its definitely better than the Moto G.

Some apps that I recommend (PRO versions = little donation / big incentive to the devs):
SuperSU, Titanium Backup, Root Explorer, 500px, Apex Launcher, Kii Keyboard, SwitchR Gestures.

As for accessorizing, these look handy:
- This case
- This memory card
- This desk/bedroom dock
- This car cradle
- This car fast-charger
- This portable Mircast dongle
- This controller to use with the Sixaxis App and play games with this mount/diy to place the phone over the controller
- OR instead, this easier but less advanced solution, The MOGA pocket / PRO controller

And if you're interested in emulators...
...here's some that run fullspeed, with sound, no frameskip, and no problems:
PS One (FPSe)
Nintendo 64 (Mupen64)
Gameboy Advance (VGBA)
SNES (Super GNES)
Sega (iGensNoid)

...and here's more emulators which (aren't finished/smooth) but might be useful:
PSP (PPSSPP)
DS Lite (DraStic)
Dreamcast (Reicast)
Windows 98/DOS (Winulator)

Anyways, enjoy your new device