attach phone to computer. drag and drop the sd card contents up to your computer.
As well, BEFORE you do this, you could even use something like MyBackup Pro and make backups of all of your apps and settings (system, phone, apps).
call logs, mms/sms can all be backed up with MyBackup Pro for example. Contacts are on the web in the cloud of course, but MyBackup Pro does those as well.
MBP does not need root.
all this being said, there is no better backup than when rooted as you can make nandroid backups which are a snapshot of EVERYTHING at that moment. bit in lieu of that, MyBackup Pro will do what you want and no root needed and then once it is done, copy your SD card over to your computer and then you have it all.
/** SGS3 (rooted & stock ROM) **/
/** Asus Prime TF201 (rooted & mildly OC'd) BRICKED BY THE ASUS OTA JB UPATE!**/
/** OG Droid (rooted & OC'd & all hacked up) **/