Apple saves your text messages in its iPhone backups—whether they're saved locally on your PC or they're part of an iCloud backup—which you should have. That's good! Unfortunately, the file ...

To save the file on your phone, simply tap on the three dots in the top right-hand corner and click on Save copy. It works in the same way from the smartphone to the computer.

The Single File browser extension can save web pages as self-contained HTML files that are readable offline, and Obsidian has a free web clipper that works in tandem with its desktop app.