There is a huge difference between an update and an upgrade. That explains "I checked my MacBook Pro for software updates and it said I didn't have any.
It sounds very much like you are running some very old version of OS X (earlier than OS X 10.8.5) right now. Please realize the current version of OS X is El Capitan (OS X 10.11). Changes from OS X 10.8 (Mountain Lion) to OS X 10.9 (Mavericks, which was released in 2013) and OS X 10.10 (Yosemite, released in 2014) are upgrades, not updates. You have to go to the App Store, and essentially "Purchase" these changes (at no charge) and download and install them.
The current version of iTunes is 12.3, and, as you say, it requires OS X 10.8.5 or later. You need a current version of iTunes to interface with your iOS device running IOS 9. Your version of OS X, whatever that is, is running the latest iTunes version it can. It has no iTunes updates available.
So you need to perform a lot of software upgrades on your computer, if the hardware can even accept them.
Upgrade to OS X El Capitan - Apple Support
Upgrade to El Capitan if you can, then upgrade iTunes.