My new IPad Mini tells me that I need Itunes 10.7 when connected to my Apple computer. When I try to up grade, I'm told that there is no newer version than what I have. IPad mini says I have system version 6.1.3. The ITunes version on the Mini is unknown.

The problem is that your Mac software is quite old. You must have a version of OS X on your Mac earlier than 10.6.8 (current is 10.8.3), so your particular version of OS X cannot support any later version of iTunes. (The current iTunes version is 11.0.2). In particular, your iTunes on your computer cannot deal with your iPad which has software that is much more current than your OS X operating system (and Mac iTunes).

You need to upgrade your computer to at least Mac OS X 10.6 to download a newer version of iTunes. See if your Mac is compatible: http://support.apple.com/kb/sp575