How to save old Messages on a Mac?
iMessage conversations appear to be saved in ~/Library/Messages.

To see your Library folder, you'll need to use Go > Go to Folder... (⇧⌘G) and then type in ~/Library. If your new Mac doesn't have any messages on it, you should be able to just copy the folder over (but make sure Messages is closed while doing it).

Make sure to back up the destination Messages folder before copying if you are worried about losing data (just rename it Messages-Old or something). If everything works okay, you can trash it afterward.