For iceberg lettuce, from a store, I make a fresh cut in the stem, put the head in a plastic bag and put a moist paper towel over the cut stem.
Some vegetables can be kept for days, even weeks. Root vegetables like carrots and beets can be kept for weeks in a cool place like a root cellar. Cabbage family vegetables can also be kept for a long time. Heads of cabbage can be kept for time measurable in weeks, and cauliflower and broccoli can be for time measurable in days.
Other vegetables are best if eaten as soon as possible after they are picked. Corn on the cob is sweetest when first picked. An old gardener's advice was to get a pot of water boiling before you go out to pick the corn. If you drop an ear on your way in from the garden, don't even stop to pick it up. Get the other ears cooking right away!
Asparagus is also best when fresh. If you buy asparagus, you can help keep it fresh by making fresh cuts in the bottoms of the stems and then putting the whole bunch in shallow water. Let it stand there and drink up the water.
Potatoes and onions do not need to be refrigerated, but they should be kept dark. A cool storage place is better for them than a hot place. They have different needs with regard to moisture. Onions should be kept in a dry place. If you live in a cool climate, you can probably keep them both for longer times than if you live in a warm climate. It is good to experiment and see what works for your situation.
Comprehensive cookbooks usually have a section about foods and how best to store them. It is worth reading these sections, especially for vegetables.
Happy cooking,
Ruth Ann