Backup your contacts to Gmail. Copy and paste pics and videos to your computer as mentioned above. If you want, you can backup SMS using apps like SMS Backup+. Unfortunately, there is no easy way to backup app data without root, which of course doesn't help in your case.
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Phone: HTC One X (AT&T version) Rooted, Bootloader Unlocked, S-Off
Tablet: Samsung Galaxy Note 8 (WiFi only) Rooted