I was given a used 3g iphone and iTunes updated the phone. It is on 4.2.1 with firmware 5.15.04.
What do I need to do to jailbreak it, so that I can download apps?
Not going to tell you how to download apps, as it's in a grey area of legality.
Jailbreaking however, is perfectly legal...
I would suggest using an app called Redsn0w. I believe the latest version should still support older phones, though I have not tried it...
There are many guides on the web for doing this. A simple Google search found quite a few. Just find one thats uses Redsn0w and your good...