You'll need kitchen tongs, a sharp knife, a fork, a cutting board, a big bowl, a plate, and a food mill.
You'll probably also want some tweezers or duct tape on hand to remove the inevitable hairy thorns that will get stuck in your skin despite your best efforts.
Step 1: Use tongs to get the prickly pears off the cactus
There are a few ways to get the fruits off the cactus. You could wear thick gardening gloves, but the problem with that is that your gloves will then be covered with insidious hair-thorns that will attack you next time you touch them. I opt to use kitchen tongs that keep me at a safe distance and can be washed clean without ever touching my skin.
Grab a prickly pear in the tongs and gently twist it off the cactus. Ripe ones are more red and will easily release from the tree. I gather a bunch of fruits in a large bowl and run the whole thing under water before the next step. 10 good size fruits will yield about a liter of juice.
Step 2: Carve the prickly pear
Be careful to use the tongs or fork the whole time - never touch the skin of a prickly pear!
Step 3: Peel the skin
Another reminder to not touch the skin. Seriously.
Step 4: Remove the seeds
There are a few ways to do this. The best way I've found is to use a simple food mill. This handy, old fashioned, mechanical device isn't used much nowadays, except by people who make their own baby food. So, if you've had a baby, or if you're just into antiquated equipment, dig out your food mill and put it to good use.
Chunk up your pile of cactus fruit and run it through the food mill. In no time flat, you'll have a bowl full of delicious pulpy juice.
Step 5: Enjoy!
(And now you might want to do a search on how to remove cactus thorns. Because, despite your best efforts, you got some in your skin, didn't you.)