Store whole potatoes in an open container at room temperature, in the shade.
Sprouting potatoes are bad, don't eat. Though if there are just tiny brown spots you can dig them out with the tip of the potato peeler.
If the skins are turning green throw them away and store in a darker place next time. Also watch for a green color around the outside when you cut them.
You don't have to peel them, but if you're making mashed potatoes you might want to. Wash the potato if you're not peeling it.
You can use a potato peeler but it's tedious. If you're boiling them anyway, might be better to boil in quarters and then peel.
Raw potato is poison. Cooked potato is soft all the way through. Check with a fork.