You can test if the tonearm and motor is gounded with a electrical continuity tester first. If they are gounded already, then no need to do it again. My old Well Tempered motor was 2 prong plug and was not gounded and made a lot of buzzing noise. I attached a wire from a screw on the metal chassis of the motor to the chassis of a grounded component and the noise went away.
Also some phono cables are really noisy and some are really quiet. You might want to play around with that also.