Ideally you would harvest right before use but when that is not possible you need to know how to store your produce. In most cases the way produce is stored in the produce section of your grocery store is how you would store it at home. The wrong type of storage can reduce the time your harvest will last and may affect the flavor.
For short term storage:
Leafy greens (including spinach, lettuce, arugula), most herbs, cabbage, asparagus, sprouts, sunchokes, berries, cherries, rhubarb
Do not refrigerate:
Onions, garlic, avocado, tomatoes, potatoes, apples, oranges, grapefruit, lemons, limes, pears, peaches.
For longer storage: root cellars, freezing, canning, dehydrating depending on the type of produce and how you plan to use it later.