Don't make it complicated
Fresh ginger has an unmistakable snap to it, so you know you've got old ginger on your hands when it kind of bends when you try to cleanly break it in two. And that's sad for everyone. So what's the best way to store fresh ginger so that it stays as fresh and snappy for as long as possible? There are actually a few different ways to store fresh ginger, and the method that's best for you and your ginger really depends on how you're planning on using this rhizome—and yes, ginger is technically a rhizome, not a root—in your kitchen.
The first step, though, is making sure you pick a quality piece of fresh ginger from the market. At its best, a piece of fresh ginger should feel firm to the touch, not soft. The outside, though wrinkly in texture, shouldn't look shriveled. And if you see mold? Avoid that ginger all costs. If you're looking for a stronger, ginger flavor, pick a piece with thicker skin; that means it's more mature and therefore had more time to develop its ginger-ness.
If you're just looking to store that giant hunk of unpeeled ginger, straight from the store, until you use it in a day or two, all you need to do is pop it in your fridge, ideally in your crisper drawer.
Ginger storage gets a little more complicated once you're sliced into it and started to peel it. But if you peeled too much ginger at once and want to store that extra nub, wrap it tightly a paper towel and put the little packet in a plastic bag. Place that plastic bag in the fridge. This paper towel-plastic bag trick also works to keep unpeeled ginger fresh, if that's something you're into.
There are others who swear by storing peeled and chopped ginger in jars full of vodka, but that could totally alter the taste of the stuff, especially if you leave it in the fridge for a while.
Really, in a perfect world, you should only peel as much ginger as you need at any one time. That way, the ginger's skin can protect it naturally, and you don't have to worry about plastic bags and paper towels and vodka.