If the cheese you bought is wrapped tightly in plastic wrap, remove the cheese from the wrap when you get home. If you bought cheese from a cheese shop and it's wrapped in wax or parchment paper, you can often just leave it in that paper for the first few days.
When storing a soft, semi-soft or semi-hard cheese, wrap the cheese in waxed or parchment paper. The paper allows the cheese to “breathe” meaning that the cheese is protected, but a little bit of air can still filter through the paper.
Soft cheese needs to have some exposure to air so it does not become too moist and watery. Wrapping a soft or semi-soft cheese in plastic wrap essentially smothers the cheese, trapping moisture that will destroy the rind and the flavor of the cheese.
Examples of soft, semi-soft and semi-hard cheeses: Robiola Rocchetta, Brie, Camembert, Blue Cheese, Manchego and other similar sheep’s milk cheeses, cheddar
If you're storing fresh cheese that doesn't have a rind, store it in an airtight container
When storing a hard cheese, trapping moisture is less of a worry. It’s more likely that the cheese will dry out too much in the refrigerator. Therefore, it should be protected from the air by being placed in a sealed plastic bag or sealed (glass or plastic) container. Hard cheese can also be wrapped in plastic wrap.
Some cheesemongers argue that this environment is too dry for hard cheeses, and that wax or parchment paper is still best.
Examples of hard cheese: Parmigiano-Reggiano, pecorino romano, aged gouda
Cheese doesn’t like to be too cold; around 53 degrees is ideal. Most home refrigerators are colder than this. As a solution, store cheese in the produce drawer, which is usually slightly warmer than the rest of the refrigerator.
If your kitchen has a cool, even temperature then it is possible to leave cheese out on the counter. However, plan to eat the cheese within a few days. You'll also want to cover the cheese with cheese cloth or a cheese dome. This will prevent the cheese from flies and from being exposed to too much air.
Freezing cheese is not recommended. If you have to, then plan to use the cheese only for cooking, as the texture won’t be ideal for straight eating.
If the cheese you’re storing becomes moldy, it’s often fine to just scrape or cut off the mold and re-wrap the cheese in fresh paper or plastic.
If a hard cheese has a white film on it or if the cheese is rock hard, it’s too dried out. Try scraping away the white film, as there might still be good cheese underneath.
If a soft cheese has a cracked or bulging rind, it just might be past its prime. Use your nose as a guide. If the cheese has an ammonia-like aroma that makes your nose hairs tingle, it probably won’t taste good.
If you have a lot of little bits of cheese in your fridge that are almost too old, then consider making fromage fort. It is a traditional French cheese spread made from several types of cheese, white wine and seasonings.