IPhone 5 set up/iTunes problem

I'm currently trying to set up my new iPhone 5. My iTunes gives the error: "You can't use this iPhone with iTunes as you need version 10.7". I have downloaded 10.7, but then that won't update because my computer says, "you need version Mac OS X 10.6.8 or later".

However, and here's the catch: the software update function on my computer says everything is up to date and there's nothing to be installed! What do I do?

Thank you!

Added (1). I have connected it to Wi-Fi, but I'm trying to do a restore via iTunes. This is my problem.

My iphone 4S is working just fine.

I would just connect the phone to my Wi-Fi instead of using the computer, it's easier that way.

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