My MacBook won't let me update Itunes to 10.7 or let me install it once downloaded from apple.com

I just activated my new IPhone 5 and went to plug it in to sync my music on to it. A message popped up saying I needed to upgrade to at least itunes 10.6 to be able to sync my phone. I went to check my updates and found that I did need to update my Itunes. (I clicked Itunes, check for updates....) but then it was not listed under what needed to be updated. So I tried updating all that was in there so maybe it would come up after but it didn't. I then downloaded the itunes 10.7 from the website to see if that would help. Once downloaded it would not install onto my computer. I do not know how I am supposed to upgrade when my computer is not letting me upgrade it! I would really like some advice as to what I need to do in order to sync my music and upgrade my itunes, thanks!