I don’t automatically peel ginger; if you choose not to, it will give the syrup a slightly darker color, which I don’t mind. If you wish to peel it, you can. The leftover ginger can be added to a batch of marmalade, or patted dry and chopped further, then added to just-churned ice cream or
gingersnap cookie dough. If you plan to reuse it, you should peel it before making the syrup with it.
This syrup is quite spicy and if you find it a bit too zippy and want to tone it down, the next time you make it (or the first time) you can blanch the chopped ginger first in boiling water, let it simmer for a couple of minutes, then drain it and proceed with the recipe.