1. Go here:
http://forum.xda-developers.com/showthread.php?t=2255652 and download. Link in the first post.
2. Extract all the contents. The main files you should have are Odin3.exe and cofface_I9500_cwm_recovery_en_new.tar.
3. Turn off your phone. Then hold Volume Down + Home + Power to boot into download mode. Your phone will prompt you to confirm you want to enter download mode. Press up to confirm.
4. Plug your phone into your PC.
5. Start Odin3.exe. If all is good, the top left block will go yellow to show a connection. Otherwise download Samsung Kies to get the drivers loaded.
6. Click the PDA button and select cofface_I9500_cwm_recovery_en_new.tar which you extracted earlier. Tick and click nothing else.
7. Click Start. Takes a few seconds and your phone will reboot.
8. Once booted, turn off again.
9. Now boot into recovery. Volume Up + Home + Power.
10. You will see a menu option "Root Phone". Use your volume buttons to get to it and power to execute it.
11. Reboot phone. Profit.
EDIT: Titanium Backup is critical if you are going to enter the ROM flashing arena. Unless you like the pain of setting everything up over and over. It is also handy to keep backups of all your data in the cloud so setting up a new phone is painless.
Other than that, do a Google for apps that require root. There are quite a few. Droidwall another handy one.