Those fresh vegetables look so good in the store! Probably because of well thought out marketing, like lighting and display, and it works even on those of us who think we're hardened against marketing ploys.
It's easy enough to buy more than we can eat before it goes bad, so it makes sense to find a way to keep them. Some vegetables are easier than others to preserve. For instance, lettuce doesn't freeze, dehydrate or can well.
However, keeping carrots for long time in the refrigerator is easy. Rinse them and let them drain. They should be moist but not wet. Put them in a plastic bag with a good closure and put them in the vegetable drawer. I have kept carrots for months like this. They can also be frozen or canned if you have enough to make it worthwhile.
Celery keeps fairly well, but don't close it up in a bag. A little moisture to the root end of a bunch seems to help. Some people say wrapping it in aluminum foil will make it last longer, but I haven't had much luck with that method.
What else? If you buy things like green beans, peas. Brussels sprouts and so on, it's best to freeze them within a few days. They go bad quickly. Blanching them will keep their flavor and texture better. There are lists and tables of blanching times for most vegetables and fruits, but I have found that two minutes for small things like peas and three minutes for larger pieces like Brussels sprouts do the trick. It's not an exact science.
There are far too many vegetables to address all in one post, but it's not hard to look up whatever you have to see what is the best way to preserve it. Not wasting the food you have is one of the best basic food saving ideas there is!