My computer says I need a newer version of iTunes on my iPhone in order to sync them but my iPhone is a brand new 7 and should already have the latest version of iTunes...help!

It is not referring to the phone, but to iTunes on the computer. iOS 10 requires the latest version of iTunes, 12.5.1 or above, which requires Windows 7 or above, or OS X 10.9.5 or above.

Are you running iTunes 12.5.1 or newer? Download update --> https://support.apple.com/HT201352

What about your computer OSX version (if you are running Windows you need Windows 7 or higher and you are posting in the wrong forum)?

iOS 10 requires iTunes 12.5.1 or newer which in turn requires OSX 10.9.5 or newer. If you use Software Update realize it only checks for updates for the current system version you are running but that itself may be out of date. It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system you will not be able to sync an iOS 10 mobile device to your present computer, though some features may be available with iCloud services such as Apple Music or Match. OSX 10.9 and 10.10 are no longer available. Check this article on El Capitan (OSX 10.11) and Sierra (OSX 10.12) upgrade options https://discussions.apple.com/docs/DOC-10692 These systems may run slower on older machines and require additional RAM purchase (4 GB is a realistic minimum and many recommend 8GB). Doing a big jump in system versions is also more likely to affect old software.

Snow leopard to el capitan, will it cause my macbook to be slow? - https://discussions.apple.com/thread/7412959

Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/

If you have any PPC generation software that has been running under OSX 10.6 it will not be supported under any later system versions.