Hakurei turnips are a member of the cabbage family. They are different than the old-school, thick-skinned turnips you may be used to and are more similar to a mild radish. The turnip itself is very white and smooth, and the greens are dark green and also edible.
The best way to store them is separate from their tops, with a sharp knife, cut the greens off close to the turnip its self, then wrap the leaves in a damp paper towel and store loosely in plastic bag in the crisper bin of your fridge. They will keep for about a week. For the turnips themselves, again, store in a plastic bag in the refrigerator or cool spot in the basement. These will keep for several weeks.
Because they are so young, hakurei turnips can be eaten raw, like a radish. They can also be steamed, boiled or roasted. The greens are also really great, and can be eaten too. Use them in any recipe that call for chard or spinach.