This may seem like a simple and self-explanatory technique, but there is definitely a right and wrong way to peel a vegetable.
First, make sure that you use a flat peeler, as shown below, not the traditional vertical peeler. A flat peeler will give you better hand position, which will allow for faster movements.
Allow the tip of the vegetable to rest on your cutting board as you hold it at a slight angle with your guide hand. Using rapid and fluid movements, run your peeler from top to bottom, spinning the vegetable as you go.
Flip the vegetable from top to bottom and continue to peel the area your guide hand was previously holding. With a little practice, your ability to quickly peel vegetables will increase greatly; which is good, because no one enjoys spending hours peeling vegetables and no chef enjoys paying you to do it.