Every iPhone user has an iCloud account which almost all are backing up to right now. So when you change phone you just need to do a Restore from iCloud and all of your messages will just be there. I would consider it pretty idiot proof.
iCloud backups are also known to fail. Someone I know just can't get the backup to work on their iPhone.
Go to the next Apple Store and enjoy a free unlimited in time support for theses issues.
Every time I came by a Genius Bar for support I have witness "dumbs" users crying tears of joy after an epic data recovery thanks to the dumb-proof iCloud backup (and that exactly why both my parents are backed up on iCloud, for some people "dumb-proofness" is more important than privacy concerns)