I'm not sure how possible it is to give good and definitive advice on how to store fruits and vegetables generally. The best ways to store food will often depend on the specific kind of food and the purpose you intend for it. So take the following advice as more of hint, as it may be very bad advice for a specific fruit or vegetable.
Generally speaking most kinds of vegetable store well in the fridge or on the shelf for few days uncut. How well they keep on the fridge or shelf after you cut them it depends a lot on the actual vegetable. For example, bell peppers seem to do OK in a bag in the fridge when cut, an onion is fine if you cover the cut side in plastic wrap, while a cut potato will dry out. You can freeze pretty much any vegetable, just check the frozen goods section of your grocery store, and safely store them frozen for years. While you probably wouldn't want to use them as a substitute for uncooked fresh vegetables (eg. in a leafy salad), they can usually be cooked just like you would as if they were fresh. You may not even need to defrost them first, often you can just substitute frozen vegetables directly.
Fruits for the most part a different matter. How best to store them, especially when cut, depends a lot on the particular fruit. Generally speaking after being cut most fruit needs to be used soon after or becomes useless for most purposes. You can freeze fruit and this will keep them from spoiling, but most fruit doesn't hold up well in the freezer. For a lot of purposes, what you get when you defrost fruit isn't a good a substitute for fresh fruit. While most vegetables will have something at least resembling their original shape and texture after being frozen, many fruits end up like mush. Frozen fruits are mainly only useful for things like filings and jam.
When freezing things, fruit or vegetables, always use an air-tight container. In the freezer air flow is your enemy. Foods exposed to the air will soon become freezer burnt. You'll want to prepare the fruits and vegetables, removing skins, seeds and other inedible or undesirable parts, as doing so after freezing will be harder. You'll also want to label them with their name and when you froze them. Since you can keep them for years, you can end up with a collection of mystery foods in your freezer if they're not labelled.
Two other possible alternative for longer term storage of fruits and vegetables are dehydration and canning. What you can do with the food after preserving them these ways is generally more limited than freezing. Dehydration often lets you eat the dried food as-is, while canning lets you preserve food in a already cooked form. You can also pickle many vegetables (and some fruits) but this essentially turns them into a different kind of food.