Open the Messages app
Find the message you want to save
Press and hold the message
Tap Copy
Paste it into Notes, Files, Mail, or another app
Take a screenshot of the message
Save the screenshot to Photos
Tap and hold a photo or attachment in the conversation
Tap Save to Photos or Save to Files
Turn on iCloud Messages to keep messages synced across devices
Go to Settings
Tap your Apple ID
Tap iCloud
Tap Messages
Turn on Sync this iPhone
Back up your iPhone to iCloud or a computer
Go to Settings
Tap your Apple ID
Tap iCloud
Tap iCloud Backup
Turn on iCloud Backup
Tap Back Up Now