rooting a phone usually wipe it entirely !
if you want to backup a phone you already used, recommending rooting is maybe not advised as he will lose everything and will have nothing to backup anymore.
of course, then he can make full rom and user data backup, but I suppose he wants only backing up his data.
some manufacturer might have tools to backup (sms, contact list, photo, etc.) to computer. if not enough, you'll have to use a program but most will need root to backup also your settings and savegames for example.
you can also use adb command line interface (it's an android developer tool) on your computer, but you have to learn how it works and which command to type to do backups of everything, or single app, etc. adb might not backup everything as some programs are protected.
there are GUI or layer to use with adb, like titanium backup.
You do not necessarily need to root to install TWRP. As long as you have unlocked your bootloader it is possible. The only problem with this is that in order to unlock bootloader your phone will wipe. I would recommend backing up with titanium using adb like @Cyan mentioned and then install TWRP. Afterwards you can restore your titanium backup and have TWRP. TWRP can be very useful especially if you somehow brick you phone or get stuck in a boot loop.