Start the peeling process by cutting off the ends of the prickly pear.
They're hard to see, but those tiny hair-like spines really are there and they'll easily find their way into your skin. To protect your hands and fingers, it's best to use a fork, a towel, or even a gardener's glove to steady the fruit while you work with it. I didn't bother doing that here so that you could see the fruit better in the photos. (And yes, I paid the price for it and spent some time with tweezers afterward to pull out the spines from my fingers!)