The vegetables and fruit that you are receiving each week will last quite some time if you provide them with the proper storage conditions. Each vegetable is different in its requirements but for an oversimplified rule of thumb, put your produce into plastic bags or tupperware in the refrigerator.

Here are some simple, but important, storage tips to help maximize the shelf life of your shares. 

Greens (i.e. salad mix, lettuce, kale, chard): Greens of all kinds should be stored in plastic bags. Do not wash them before storage as water sitting on the leaves will cause them to rot faster. 

Root Crops (i.e. carrots, beets, potatoes, radishes): If there are tops on your root veggies (like on a bunch of carrots), take them off before storing. Simply snap or twist off the tops as close to the root as you can. Leaving the tops on creates rubbery vegetables very quickly as the tops suck the moisture out of the roots in an effort to avoid wilting. Store root crops in plastic bags in the refrigerator, ideally in the crisper. The other option is to store them in a cool, dark place with temperatures as close to freezing as possible. Many times the corners of basements work well. If you do not have enough space for everything, potatoes will keep reasonably well for a few weeks placed in paper bags in a cool, dark space such as a cabinet. 

Tomatoes: Tomatoes should be stored out of the refrigerator for best flavor/maximum enjoyment. 

Herbs: Most herbs should simply be placed in a plastic bag and put in the refrigerator. Basil is the exception as it will taste best if it is not refrigerated. When you have an entire basil plant, including the roots, just stick it in water like a bouquet and pick leaves off as you need them. 

Corn: The sugars in corn begin converting to starches the second it is picked. The sooner you eat the corn in your share, the tastier it will be. If you are going to let it sit in the refrigerator for more than a couple of days, to actually enjoy the corn your best bet is to take the time to freeze it for later use. 

Fruit: Berries should be stored unwashed in a tupperware container and refrigerated. Apples will retain maximum crispness when kept at temperatures around 32°F. However, apples produce ethylene gas, which causes off-flavors in many different vegetables during long-term storage. Keep apples in the refrigerator, preferably the crisper, and in separate plastic bags. 

Alliums (onions, garlic, shallots): The first onions and garlic you receive in your shares will be fresh (not cured) and need to be kept in the refrigerator. When the onions and garlic have dry skins on the outside, they can then be kept in a dark, cool place in your house. Darkness is very important to prevent sprouting. Unheated basements, cabinets, and refrigerators are all good options when storing alliums. 

Cabbage and Brussels Sprouts: Both of these should be stored in plastic bags in the refrigerator. Brussels sprouts should be stored after they have been removed from the stalk. If there is spotting or yellowing on the outer leaves of your cabbage or sprouts, as long as they are still firm you can simply peel off the outer layers before cooking and they will be good as new. Cabbage also stores well even after it has been cut, just trim off the exposed edge before you use it next. 

Winter squash Squash keep best at 55°F. The key to long-term household storage is to find a cool place with stable ambient temperatures and low humidity. If you have more than a few squash stored together, it is worth checking them all for rot every couple weeks to make sure a rotting squash does not spoil the pile.

Mushrooms: Remove the mushrooms from the plastic bag they are delivered in and place them in a paper bag in the refrigerator. This dry environment will greatly extend the shelf life of your mushrooms. 

If you want even more details about specific vegetables, you can find storage tips on a crop-by-crop basis by visiting Angelic Organics.