Drag the vegetable peeler slowly across the skin of the orange. Start at the top, near the navel, and slowly rotate the orange as you peel the skin. Try to remove the skin as if it were a long piece of fabric covering the ball. If you can remove the skin in a long strip, it will make later steps easier.
Separate the pith. Lay the peel on a flat surface, white side up. Lay the knife flat against the peel, with the flat part of the blade parallel to the cutting surface. In a slicing motion, carve the pith from the zest. A long peel will be easier in this situation. After you have cut the large sections of pith, go back and trim the small sections the knife may have missed.
Cut orange peels to size. Slice and dice the zest according to your needs.