If you have a mac you bring up imessages on there, scroll up to the point where you want to start the record of messages, then click file, then print.

This will save all the imessages but wont include the date and time for every message. It’ll only show the date and time that pops up in the middle of the conversation every once in awhile.

The only other way after that that I am aware of would be to take screenshots on your phone.

@Heather Mack here’s search results on Apple’s support forum for “iMessage export: https://discussions.apple.com/search?q=imessage%20export.  Perhaps you could find something there that might help?  (I’m not an Apple user so can’t offer any advice beyond suggesting to search the Apple community for answers.)