Go to Settings 
> [yourname]> iCloud, then turn on Messages (if it’s not already turned on).
Every message you send and receive on your iPodtouch is saved in iCloud. And, when you sign in with the same AppleID on a new device that also has Messages in iCloud turned on, all your conversations show up there automatically.
Because your messages and any attachments are stored in iCloud, you may have more free space on your iPodtouch when you need it. Message bubbles, whole conversations, and attachments you delete from iPodtouch are also deleted from your other Apple devices (iOS11.4, iPadOS13, macOS10.13.5, or later) where Messages in iCloud is turned on.
See the Apple Support article Use Messages in iCloud.
Note: Messages in iCloud uses iCloud storage. See Manage Apple ID settings on iPodtouch for information about iCloud storage.
