Two days ago, I had my cartilage pierced with a gun. The stud is able to turn and I'm cleaning it but it still throbs a little. I don't believe it's infected but when should I expect the initial pain to go away?

Anywhere from a few minutes to three months.

STOP TURNING YOU JEWELRY. This is an old wives tale from when they pierced with gold. It's doing more damage that good and could be why you're still throbbing a little. It only takes a little tiny piece of crustie to rip open a freshly healing piercing.

Hopefully you're ear is all intact inside and the pain will go away in about two weeks total. (Not really pain after the first three days, it's discomfort from having a healing wound on your head... but same difference for here)

My pain was gone inside of a week, 2 weeks is the average, and it will probably be tender to lay or put pressure on for months to come.
And yeah, braleighb 13 is right about the turning, Besides it turns on its own from just day to day life, right? :P

Oh! Almost forgot most important bit, if you ever get another cartilage piercing, get it done with a needle, it doesn't hurt any more (I found it hurt a lot less actually) but it is hugely safer, as its less likely to shatter your cartilage!