Flashing CM 11 on Verizon Samsung Galaxy S3 (SCH-I535)
Too lazy to post a detailed answer. But no, you need don't necessarily need root to install a custom ROM like CyanogenMod. But, you need to flash a custom recovery like ClockworkMod (CWM) recovery which you will use to install CyanogenMod. Some custom ROMs (like CM) will give you root.

For detailed installation guide for your phone, refer to (CAUTION: This is for CM 10.2, but the steps should be similar for CM 11) CM's official wiki . It contains all the information you need, including back-up.