OnSong Pro is purchased on your Apple ID so as long as you are using the same Apple ID, you can use it on any device you own. This means you can sign in with the same Apple ID and download OnSong Pro again for free. Keep in mind that you may need to sign into the App Store so that the OnSong information screen shows a download button instead of a price.
If you are using a different Apple ID but purchased it through a family member, you may need to set up iTunes Family Sharing. This would let you share apps you purchase on one account with another family member as long as both accounts share a credit card. Here is information on how to set that up: https://support.apple.com/en-us/HT201088
You can easily transfer your library from one device to another by backing up and restoring the backup file. Here is a brief tutorial that shows you how this is done: http://onsongapp.com/videos/transfer-dropbox
If you don't have a Dropbox account, you can also transfer your library using iTunes File Sharing: http://onsongapp.com/videos/transfer/itunes
If you previously had purchased legacy add-ons in OnSong Pro like OnSong Console, Drawing, or other features, you may need to restore those purchases on the new device. To do this, go into the Utilities Menu by tapping on the gear icon in the Menubar. Next, tap on the Account option and choose the Add-ons option. Tap on the Restore Past Purchases button at the bottom to have OnSong Pro request purchased from your Apple ID and activate those features in OnSong Pro.