For the freshest Ginger only peel off what you need for your recipe. You can store unpeeled Ginger in the fridge in a tightly sealed plastic bag. You can also store it in a paper towel then place in a paper bag. There are other methods for freezing it as well.
My preferred way to store ginger is to wrap it in a dry paper towel then place into a sealed plastic bag, then place it into a green veggie bag and store in your veggie drawer of your fridge. I still have some left over from Thanksgiving that is fresh.
See more ways to store and freeze ginger at wikiHow.