If you're mostly interested in keeping your messages around somewhere (without necessarily having local access to them), then SMS Backup+ is probably your best choice. The app is open source and allows you to automatically back up your SMS, MMS, and call history to your Gmail account, presenting your messages there in a convenient "SMS" label, laid out in the same format as email conversations, accessible anywhere via phone, computer, or tablet.
The setup process is simple. Once you have the app installed, fire it up, and jump past the changelog splash screen to the main app. There you toggle the "Connect" switch, grant the app access to your contacts, select the Gmail account you'd like to have your messages synced to, and decide if you'd like all your current messages to be backed up or not as it pulls SMS permissions. That's pretty much all there is to it.
You can set the app to automatically back up messages with a configurable schedule. I've always found the default settings—which back up incoming messages every minute and outgoing messages every two hours—to be fine. Data used by the app is pretty minimal, but if you are concerned, you can set it to only make backups on Wi-Fi. It's also able to broadcast an intent at the time of backup for third-party app integration.