What is the bare minimum that needs to be done to titanium backup to work? I am not big on flashing a custom rom at this time. Do I still need to install a TWRP? Or can I just unlock the bootloader and install supersu?
Bare minimum is rooting, which can be done by methods suggested for your device either using PC or one click apps. Search for methods for your device in rooting on this site or on XDA forums with particular attention to your specific device model and carrier (if applicable)
Unlocking bootloader does NOT equal rooting. That is usually the first step in rooting process. Once your phone is rooted SuperSU is a kind of permission manager to give root access to apps of your choice
You don't need to install TWRP or a custom recovery for your purpose but it would help you in taking complete backups ( Nandroid backups of your device ), even if you don't wish to install custom ROMs