There are important data within the texts and multimedia items of a conversation that were sent and received on my LG Nexus 4 phone. I would like to have an backup and preserve of them on Android device, like PhoneView for iPhone.
You could try the following apps in Google Play store to do what you need.
* MyBackup Pro offers the easiest way to backup Contacts, Call log, SMS (text messages), MMS (message attachments) and others to your SD card or online remote server.
* Titanium Backup (Rooted Users). You could create label and check "[SMS/MMS/APN] Dialer Storage 2.2 with the app, then it should automatically backup the SMS/MMS and Call Log.
* SMS Backup & Restore is a free app for getting all text messages into XML format, but doesn't support mms.