How do you pickle pickles?
This is a very vague question--there are many ways: quick pickling; canning; brining; fermenting...For starters, I highly encourage you to find a good, reliable book on canning. Ball makes a great one--I think it's called the Ball Complete Book of Home Preserving. It covers all the bases, and you'll want to brush up on canning safety information.
If you want to make quick pickles--i.e. pickles that are refrigerated and thus do not need to be sealed in a jar--you have more leeway. Basically, a pickle can range from vegetables that are shredded or thinly sliced and marinated in a vinegar-based solution, to the pickles we are most common with, such as dill pickles or pickled okra.
For quick pickles, slice your vegetable of choice very thinly (carrots, cucumbers, cauliflower, parsnips...the list goes on and on--you can pickle almost any vegetable), and throw them in a jar with a solution made of the vinegar of your choice (apple cider, white, red wine, white wine, rice, etc.), a little water to tone down the vinegar, a little sugar, and any spices you like (cloves, star anise, cinnamon, peppercorns, and mustard seeds are very typical). You may also want to bring the liquid to a boil, and steep the spices in it before pouring it over your veggies.