iOS 10 requires iTunes 12.5.1 or newer which in turn requires OSX 10.9.5 or newer. 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 this phone 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.
The option to roll back to iOS only exists a day or two after a new iOS is released and iOS 10 was released more than a week ago.
