Dry enough to grate
Parmigiano-Reggiano, Grana Padano, Pecorino Romano, and an aged Gouda grate cleanly. They are low in moisture. A young, moist cheese clumps on the grater. Slice it or melt it instead.
Grate it cold
Cold cheese makes smaller pieces and sticks less. Use a fine grater to finish a hot dish. Use larger shreds when the cheese needs to melt into the food. Grate what you will use, then wrap the cut face again.
Where it goes
A grating cheese is often a finish, not the base of the dish. Add a hard cheese off a hard boil so the fat does not split. These cheeses are salty. Taste the dish before you add more salt.