I am not sure how to do it on the device, but I use SkyDeck to save all of my text messages. It is a free service and has a few extra cool features. The caller ID is kind of neat and it also keeps all of your contacts on line as an added back up.
I think it is secure. I have been using it for months now with no issues. All of my texts and contacts are saved and there are multiple options for keeping information private. It has been a lifesaver for me when I have to go back to a text from weeks or months ago.
