What is the easiest way to upgrade my Galaxy S2 to Android 4.2.2 Jelly Bean? Also, how do I update it?
Here's some ways to update your firmware, from easiest to harder (not necessarily hardest).
OTA (Over the air) update. Trigger it via
Settings > About > Software update.
Kies. Install Kies on your PC, run Kies, and connect your mobile phone to your PC using USB cable. There is an update function in the Kies software.
Odin (Windows). This is a 3rd-party tool that you use to flash (or install, in layman's terms) a stock (or default) ROM or custom ROM on your phone. You will need to download Odin and the custom ROM or stock ROM from somewhere. There are several guides on Google on how to flash a stock / custom ROM for Samsung Galaxy S2. For Linux, you can use Heimdall.