I store almost all of my produce in plastic bags or airtight containers in the vegetable drawer in the fridge. The produce I have successfully stored this way includes peaches, nectarines, pluots, apricots, cucumbers, radishes, carrots, bell peppers, onions (yellow, purple, and scallions), raspberries, blueberries, strawberries, eggplant, zucchini, melon, lettuce, celery, cabbage, fennel bulb, swiss chard, kale, lemons, limes, beans, and snap peas. Everything I store in the fridge keeps for at least week, most of it keeps for two weeks and some of it (beans, chard, radishes, kale, celery, fennel bulb) will keep for longer. I've even managed to keep zucchini this way for over two weeks (but it did go straight from the garden into the fridge).
There are only a few exceptions:
- Tomatoes: I usually store tomatoes in a bowl on the counter (out of direct sunlight). I'm not sure your idea of wrapping them in newspaper will work. Tomatoes give off a gas that causes them and other fruit to ripen. For example, if you have some underipe peaches, you can put them in a paper bag with some tomatoes, close up the bag to trap the gas that the tomatoes give off, and the peaches will ripen in a day or so. If you wrap the tomatoes in paper, this will trap the gas that they give off, causing them to ripen further. If they are underipe, that's exactly what you want. But if they are already ripe, you don't really want them to ripen anymore. I generally store my tomatoes out in the open (but not in direct sunlight). If they are really ripe, I put them in the fridge, but that does make them less palatable for eating sliced (although they will still be fine to cook with). It would probably be good to store them in a cool dark place, but I don't think I'd wrap them in newspaper. However you store them, I think you'll find that they don't last long and that it is best to try to use them within a few days. Tomatoes are fragile.
- Potatoes: I usually store potatoes in the cupboard (i.e., a dark, cool place) but I will put them in the fridge if I'm not going to use them within a week.
- Garlic, shallots, and ginger root: I store garlic, shallots, and ginger root in a bowl on the counter.
- Mushrooms: Probably best stored on the counter in a plastic bag, but really should be used within a few days or they will get mushy.
- Onions: These can be stored on the counter or in a dark cupboard, but I think you'll find they are less pungent (i.e., your eyes will water less when you cut them up) if you store them in the fridge.
- Limes and lemons: These will also do fine on the counter or in the cupboard for probably a week, but any longer than that and you should put them in the fridge.
You definitely want to put any produce that you store in the fridge in an airtight container or plastic bag, otherwise it will dry out very quickly (sometimes in just a few hours). I've read that it is best to use a perforated plastic bag (there are special vegetable storage bags you can buy) but I've never done this.