You can't save an entire chat in Teams yet, but here a couple of workarounds.
Select the chat to capture and do one of the following:
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Select the text in the message and press Ctrl+C. Paste it into Notepad or a Word document. Since you can't always select all the text, you may need to do several selects to get what you want.
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Using Teams for the Web, open the chat and right-click the message. Choose Print in the browser or press Ctrl+P and print it to a PDF. You can only print one page of a message at a time.
For info on creating a favorite of a message, see Save a message.