SMS Backup + isn't the only option for getting your texts and call data off your Android smartphone, but it's one of the simplest and most effective. It archives all of your messages to Gmail (or Google Calendar), and it covers WhatsApp communications too. The app is free to download and install, though you can leave a donation for the developer if you wish.
You'll need to activate IMAP in Gmail so the app can upload your messages to it. With this done, setup only takes a few taps -- you can assign a custom Gmail label to SMS messages, as well as limit the backup to texts from a certain group of contacts if necessary. You can run backups manually or on an automatic schedule.
If you don't want to have your usual inbox routine interrupted, there's the option to automatically mark messages as read on arrival, and you can create an auto-archive filter from the Gmail end too. SMS Backup + is also capable of restoring texts, should your phone disappear and all your messages with it.
There's no option to reply to texts from your inbox, so why bother? Having an archive of your texts and picture messages is useful in all kinds of situations: checking addresses and phone numbers you've been sent over SMS, keeping hold of conversations that hold sentimental value, searching for dates and directions, and so on. It also means you can clear out your default SMS app on a regular basis, safe in the knowledge your messages are stored in the cloud. [SMS Backup]