There are any number of reasons that you may want to save an entire Message thread - perhaps you will need it for legal purposes, maybe you want to keep it for memories. Whatever the case may be, there are a few methods you can employ to save snippets of text conversations or entire conversations if they are relatively short. Unbelievably, however, Apple has not developed a simple way to export a long text conversation - your best bet is to use third-party software like Chatology.
For snippets of a conversation you can simply email them to yourself (or export another way) by long pressing on the text bubble that you want to save and select "More."
Now tap on the circles next to every bubble you want to save and then tap the arrow in the bottom right corner.
Another way you can save message threads is through the app on your Mac. Open it up, go to the thread you want to save and then go to File -> Print.
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Now choose Save as PDF from the dropdown menu at the bottom left.
The problem here is that it will save from wherever you scroll back to in the conversation forward. So saving many years of a conversation would require a lot of scrolling back.
If you have a really long conversation you will want to use third-party software like Chatology. After downloading the app and giving it access to your Messages, click on the conversation you want to save, then go to the File menu and click "Export all conversations with (name)." Now just choose a location and click save.
You will now have a folder filled with files for each day of conversation.
Chatology offers a 14 day free trial period after which you will have to pay $19.99 to continue using it.