There’s so much waste when you peel fresh ginger, and it’s really not necessary – either scrape off the outer skin, or just chop it up peel and all
I used to peel my ginger badly, trimming off chunks of skin until I was left with a hexagonal prism and throwing a good third of the original unpeeled knob in the bin. Then a friend gave me this brilliant tip: instead of using a peeler, simply rub off the outer layer of skin with the back of a knife or spoon. These days, however, I rarely peel ginger at all; instead, I grate or finely chop it skin on, so now I waste nothing and get the benefits from the extra fibre, too. The skin is so papery, it all but disappears as it cooks, anyway.
In my experience, ginger keeps best unwrapped at room temperature in a dry place out of direct sunlight. That way, instead of moulding, it slowly dries out, preserving it for ever more (if you do want to store it in the fridge, it keeps best in an airtight container). If you have an abundance of ginger, you can preserve it by finely chopping it, freezing it on a plate, then crumbling it into a plastic tub and storing in the freezer for as long as is needed. Alternatively, preserve it by pickling it sushi-style.