If you are like me, you held off upgrading Notes while upgrading your iOS devices to iOS 9. That's because all the great new features also meant notes in a new format that wouldn't sync with the Notes app on your Mac ... until OS X Capitan.
So if you upgraded to OS X El Capitan today, it's now safe to upgrade Notes on iOS. You do so by pressing "upgrade" within the Notes app on iOS.
While there, it's a good idea to check for any notes under a section titled "On My iPhone" and to move those to the iCloud section (assuming you want them to sync via iCloud).
Related: To learn about the new features, check out Apple's page What’s new in iOS and scroll down to the section on Notes.