Cheese is best stored in the refrigerator as close to the bottom of the appliance as possible - the vegetable compartment is ideal. Wrap it in aluminum foil or waxed paper (waxed paper being the best of the two). Wrapped this way, the cheese will be able to breathe and further develop without drying out. Cheeses of various types need not be segregated. They may share the space inside the compartment.
The harder the cheese, the longer it will stay fresh (up to a month) in the vegetable compartment of the refrigerator. Semisoft cheeses will usually be fine for several weeks. Cut pieces of soft cheese will last for less than two weeks. Cheese suffers enormously from being frozen, so simply get the notion out of your head.
When wrapping it, the main thing is to be sure the cheese is kept covered and moist, still able to breathe, but without breathing so much that it dries out and cracks. To accomplish this, wrap the cheese tightly in aluminum foil or waxed paper. (If the cheese weighs more than 3 pounds (1 1/2 kg), fist wrap it in damp cheesecloth or other cloth and then in foil.)
Though at one time I railed against it, my feelings about plastic wrap have tempered over time, as the quality and consistency of this irreplaceable kitchen material have improved. While I once loathed allowing a petroleum product to touch any food - certainly not my hallowed cheese - I now discover that after all my ranting and raving about it, plastic wrap really does a pretty good job. I must offer one caveat, however: If the plastic wrap is in contact with the rind or interior of the cheese for too long (more than a week), the cheese begins to suffer; it suffocates, becomes slimy-rinded, discolored, and certainly less than it should be, or was the week before. But guess what? Had you used aluminum foil, cheesecloth, waxed paper, or a combination of all three, you would have fared no better. The rule of thumb is that harder cheese will last longer than softer ones, with semisoft (or semifirm) cheeses falling somewhere in between.
Whether you use aluminum foil, waxed paper, or plastic wrap, be aware that once you've unwrapped a cheese stored in the refrigerator, the wrapping should be discarded and the cheese should be put in a new wrapping. Reused materials won't reseal properly.