Copying messages is easy. The trick is to do it in a format that you can read on a device other than another android phone.
Any message backup app can be used to transfer to another phone. Many of them back up to XML files, so in principle you can read them with a text editor (but the formatting won't be nice). Or you could install SMS Backup+ on the phone and let it copy them all to a subfolder of your GMail account (which makes them readable by logging into GMail using a web browser, but won't structure them as conversations).
There may be a message backup app out there which has a PC companion app, but I can't name one myself. Manufacturers such as Samsung may have their own apps, but the drawback with something manufacturer-specific is that it's manufacturer-specific.