Just about anything that is vaguely fiberlike can be spun. The most common fibers used for handspinning are sheep's wool, cotton, silk, alpaca, mohair (from angora goats), and angora (from angora rabbits). In my opinion, the easiest to learn to spin is sheep's wool, although the principles are the same for spinning any fiber.
You'll want to work with prepared sheep's wool, known as roving, which has been washed to remove excess lanolin (grease) and carded or combed to orient all the individual fibers in the same direction. You can find roving at your friendly local yarn store, or from several online retailers. I like to peruse etsy.com for handpainted roving, myself. Try to avoid "top" for now, which is wool that has been combed to leave only the longest fibers, which makes things more difficult for the beginner, although it is delightful to spin with a little practice under your belt!
The wool pictured in this instructable is a Romney and Merino blend (these are breeds of sheep.)
At it's simplest, a spindle is really just a stick with a weight on it somewhere (aka a whorl.) Drop spindles come in two main flavors: top whorl, and bottom whorl, cleverly named for the location of the whorl on the spindle. Spindles also come in many different weights, but I recommend starting with a spindle that is around 2-3 ounces.
The spindle pictured is a top whorl 1.5 oz spindle, although you could also use a bottom whorl, or even the spindle that stores your CDRs. Here is an instructable that will show you how to make a spindle from a CDs, a dowel, and a hook: http://www.instructables.com/id/Drop_Spindle_Constrution
. You can also find spindles from online retailers or at your local yarn store.