I currently have ios 10.8.5 on my Macbook Air. I need 10.10.5 to download the newer iTunes so I can back up my iPhone 6 . What is the easiest way to do this?

There is only one way. Open the Mac App Store and download the macOS upgrade that is available for your computer.

See here: Upgrade to macOS High Sierra - Official Apple Support

if your Mac does not support macOS High Sierra, you can upgrade to El Capitan: How to download OS X El Capitan - Apple Support

Check to make sure your applications are compatible.

Application Compatibility

El Capitan 10.11 Compatibility information

macOS Sierra 10.12: Compatible Apps

Also check to make sure there is a compatible driver for your printer.

Do a backup before installing, preferable 2 backups on 2 different drives.

I currently have ios 10.8.5 on my Macbook Air.

iOS is for iPhones and iPads.

There have been many improvements in macOS since Mountain Lion, especially with the introduction of Mavericks. What haven't you been upgrading?