When you buy fresh ginger, it looks like the knobby alien structure above. Not exactly easy to work with.
So, when I get it home, I break the whole structure down into roughly 1inch square chunks, or the size of a very large clove of garlic. You don't want to make the pieces too small because remember you'll have to hold on to it while grating it later.
Clearly images are not of what I actually do, but a decent visual guide.
Some people will say you can peel ginger with a spoon, others a regular peeler, but I just use the knife I'm chopping it up with. Use whatever method you are comfortable with.
Finally, once I have a pile of peeled ginger nuggets, I double-Ziploc-bag them and store them in the freezer (label the bag!) so the ginger lasts for months and is even easier to grate when removed frozen.
Hope that helps!
Also, for those who like ginger tea or suffer from nausea, you can easily remove one of your frozen pieces, slice it into finer discs, and simply add them to hot water for a tummy-soothing (and sinus-clearing) brew. It is also said that fresh ginger can help ward off the worst of your pollen-allergies, tho I don't know if that is from the sharpness of scent or an internal, metabolic reaction.
Try it out!