Apple provides you the method to transfer all content from on iOS device to another. Transfer content from an iPhone, iPad, or iPod touch to a new device - Apple Support. However, not suitable for transfer specific items.
If you want only your notes or music to your new iPhone.
>>>>>Notes:
Make sure both of your iPhones Setting > iCloud > Notes is on, and use the same ID.
>>>>>Music:
Method 1, Open iTunes on your computer, and sync your iPhones. (Same Apple ID)
Method 2, Home Sharing (iOS 8.4 now don't support home sharing for music, make sure -8.4)
Method 3, Directly download songs from iTunes by using the same ID.
Method 4, Use third-part tool to transfer between two iPhones directly. (Available for different ID)
Resources:
Transfer songs iPhone to iPhone - iMobie Guide
Copy Songs from One iPhone to Another iPhone - Apple Support