Store fresh cheese in the coldest part of your fridge. This is typically furthest to the back in the main compartment. This family includes cheeses like mozzarella, ricotta, chevre, and cream cheese. All other cheeses should be stored in your produce drawer, the warmest and moistest place in your fridge.
Every time you use a cheese, rewrap it in fresh plastic. As a good rule of thumb to maintain freshness, only buy how much cheese you’ll use in a week.