I drink ginger tea many times a day, so I purchase large pieces of ginger. They always have nubs, bumps and curves, as they are actually the root, or rhizome, of the ginger plant, and that's how they grow. I don't have one on hand at the moment or I'd do a pictorial demonstration.
I call the sections that jut out from the large piece "fingers" because the big piece sometimes looks like a hand. I have no idea what the technical term is! Sometimes before peeling I break off some of those fingers, especially if they're very tight with little or no ability to get in between them. My peeling method is the same, I just do more separate parts at a time.
My favorite way by far to peel ginger is by scraping it with the edge of a spoon. It works in a similar way as a carrot peeler or paring knife, but is much easier because the spoon slides over and around the nubby parts, and gets into those v-shaped bends. Also, it takes off just the thin outer skin while leaving you with the most amount of ginger.
I'll explain one method, and once you see how easy it is you can customize the same basic idea, without needing to strictly follow my directions.
Hold the spoon with the side against the ginger, with the inside of the round part facing you. Tip the spoon gently on its edge and scrape towards yourself in such a way as to gather the peel inside the spoon. Clean off the spoon with your hand once the skin covers the spoon or gets in your way. You can scrape with the spoon faced the other way, but I find it harder to handle and not as efficient.
You can do short, quick motions, or long scrapes if there are large sections that aren't bumpy.
When I'm all done, I'll use my fingernails to get at any left over cracks and crevices. You don't have to worry about leaving those bits. As others have said, peeling isn't even necessary, but I'm kind of obsessive about it!
As for the black spots, I don't know if there's a general rule, so I don't want to steer you in a wrong direction, but I'm with JoseNunoFerreira in terms of removing them. Many times they're just dirt, as ginger is grown underground and, at least in the United States, undergoes minimal handling on the way to the market. If I can rub the black off easily with my finger, or if it smells like the earth, I feel pretty sure that it's dirt. I cut that section off to the point where all the black is gone, and use the rest.
However, if there's any trace of mold anywhere, I throw the whole piece out. A lot of people I know just cut off the moldy areas, but I always err on the side of what I know to be safe, which to me means no mold!
I also remove those nubs, or any other area, including the ends, if they're shriveled. While not necessarily unsafe, it means they're dried out and won't make yummy tasting tea anyway. They're also hard to peel. Just cut or break those off.
I agree with Dorothy that frozen ginger can be much easier to peel. I don't freeze mine very often, but when I do, I use the same spoon method for peeling. That's just my preference, because I'm less coordinated and not very good at handling a microplane! Perhaps that could be a trial and error thing for you.