A plastic protector will protect against scratches, and give you just a bit of protection against breakage.
A tempered glass protector will protect against scratches (to the screen, not to the protector) - but if the screen is hit, the best the protector can do is prevent the glass (the digitizer) from flying all over the place - tempered glass crumbles. And it's no harder than the screen.
So, for my money, either a case with a built-in protector (Otterbox, Lifeproof) or a plastic 3-for-$10 protector. (I've had one on one phone and one on the other phone for years.)