iTunes won't update, so I can't restore my new phone

The latest version of iTunes is 12.3.2, and requires a later version of OS X than you have listed. See these Mac Requirements. You will need to update your OS X.

Mac System Requirements

Hardware:

• Mac computer with an Intel processor
• To play 720p HD video, an iTunes LP, or iTunes Extras, a 2.0GHz Intel Core 2 Duo or faster processor is required
• To play 1080p HD video, a 2.4GHz Intel Core 2 Duo or faster processor and 2GB of RAM is required
• Screen resolution of 1024x768 or greater; 1280x800 or greater is required to play an iTunes LP or iTunes Extras
• Broadband Internet connection to use Apple Music, the iTunes Store, and iTunes Extras
• Apple combo drive or SuperDrive to create audio, MP3, or back-up CDs; some non-Apple CD-RW recorders may also work. Songs from the Apple Music catalog cannot be burned to a CD.

Software:

• OS X version 10.8.5 or later, Apple Music requires OS X version 10.9.5 or later
• 400MB of available disk space
• Apple Music, Apple Music Radio, iTunes in the Cloud, and iTunes Match availability may vary by country
• iTunes Extras require OS X version 10.10.3 or later

You need to be running OS X 10.8 or higher to be able to download the iTunes version you need which is iTunes 12.3 or higher.

Best of luck,

GB