What’s the Best Way to Save Text Messages from an iPhone?

For future reference, you might want to look into Smarsh Professional Archive; it’s a solid option for archiving messages.

Don't forget that both SMS and iMessage messages are automatically backed up with iCloud, so that could be an easy option too.

I recommend using iMazing. It's a bit on the pricey side, but it's excellent for managing multiple devices and archiving messages, especially if you need them for legal or compliance reasons.

If this is for regulatory compliance, I recommend not using SMS for business communication. It's best to go for official channels whenever possible.

If you're saving messages for future reference, maybe consider using iMessage Exporter from GitHub. It allows you to export your messages as both text and HTML files, which is handy!