I use super backup pro to save and transfer my SMS, contacts, calendar, and call logs from phone to phone. You also can send the backup files to your email to reinstall in case you need to do a reset of your phone.
They have 3rd party apps that can export/import SMS messages. I've done it in the past.
BTW, the default setting on the messaging app limits up to 200 messages before it starts deleting it. You can raise that limit higher. From past experiences, very long SMS threads slows down the phone and takes up RAM.