First install the app on your new device, then use one of the methods below.
Method 1: using cloud synchronization
If you use synchronization, you can simply log into your account and your data will be restored from the cloud.
Method 2: using the migration assistant
Without synchronization, the easiest way is to use the automatic migration tool. On both devices, go to Settings » Migrate to new device. Follow the on-screen instructions. Make sure to choose “old device” on your current device and “new device” on the device you’d like to migrate to.
Method 3: using a backup
If you have a backup created with Notes you can simply copy/email that to your new device and import it there. See Importing a backup for more information.