How to keep cut vegetables fresh

The best way to store vegetables is with the skin, but if you want to keep cut vegetables fresh than make sure that you store it in an air tight container in your refrigerator. If you are slicing a tomato or onion then place a paper towel on it. Store any vegetable away from apple because it gives off ethylene gas that spoils other things. Separate vegetables in different containers while storing it and use branded containers like Tupperware. Do not wash your vegetables just before storing it; wash it when you use it.
If you are storing out of the refrigerator then protect it from direct sunlight.

@Lakshya
There are several ways to keep cut vegetables. One can use the food storage bags, Tupperware Fridge Smart containers and more. Green bags are also useful, but one must make sure to keep the moisture out of the bag. Here are few suggestions
*The number one consideration should be to keep cut up foods away from apples as they give off ethylene gases and those gases are known to promote spoilage quickly.
*Left over salad can be kept in a clean dish towel. Fold the dish towel over so that none of the salad is exposed. Then place the wrapped salad in something like one of the Green Bags and remove the air before sealing it. Salad that has been wrapped in this manner can be kept for as long as 3 or 4 days.
*Broccoli can be cut up in advance, but its best not to wash it because the moisture will cause the flower head to soften and start to go bad. Cut up broccoli florets when stored in an air tight container can be kept for four to five days or longer.
*After slicing a tomato or onion, place a folded paper towel over the cut edge and either wrap in plastic or place in zippered sandwich bag.
*Celery stored in ice water in a Tupperware container lasts for at least a week.
The important thing about keeping vegetables is to be sure that they are in air tight and moisture proof containers, and that before they are stored; you have taken measures to ensure that any residual moisture from the washing process has dried.