Cryovac (or the equivalent) is the way to go, if you have the equipment. Failing that,
very tightly wrapped plastic film will work. Air is the enemy of cut, stored cheese.
I buy Parmigiano and Peccorino typically in 350 to 700 gram chunks, fresh cut and vacuum packed by the merchant. Anything I don't intend to use quickly gets wrapped in waxed paper and then in plastic film. That stays fresh in the fridge for several weeks - unless I eat it first
Do not freeze any cheese that you intend to eat (as opposed to combining in a cooked dish) - it completely destroys the texture.