Besides your spindle, you're going to need some wool. Even if you don't intend to spin anything but mohair or alpaca, you should learn to spin with wool. It's much easier and much cheaper. Once you're comfortable with the basics you can experiment with other fibers.
Buy prepared wool ready to spin for your first tries, too. Later on you can work with fleeces right off the sheep if you like, but for now you just want to get a feel for the basic activity of spinning. Roving is the most common form of prepared wool; it resembles a rope of fiber. While roving comes in many types, such as finger roving, pencil roving, and sliver (pronounced sly-ver), the type is not important.
You'll soon learn that all wool is not the same. Different breeds of sheep produce different types of wool--sometimes incredibly different. As a beginner, start with wool from a breed such as Coopworth or Romney.
The last thing you'll need is about two feet of store-bought yarn--yeah, that stuff you're going to want to throw away once you can make your own!
Getting Started
Before you can do anything, you need to attach a leader to your spindle. A leader is simply a piece of already spun yarn--in this case, store-bought--that you tie to your spindle and attach the wool to. Tie one end of your store-bought yarn underneath the whorl. If you have a bottom whorl spindle, the leader will be tied on the short piece of shaft; if you have a top whorl, the leader will be tied to the long piece of shaft. Once it's tied securely (use a square knot), bring the other end of the leader over the whorl and through the hook at the top. If you have a spindle with a notch instead of a hook, make a half-hitch around the notch. Now tie a slip knot at the very end of the yarn.
Your spindle is prepared. Time to look at your wool. You probably have fairly thick roving, so pull off a piece several feet long and pull it into at least two strips. When the strips are the right thickness they won't seem densely packed, but they'll still hold together well.
Tease out the end of one of the strips and string it through the slipknot on the end of your leader, tightening the loop onto the wool. Double the fiber over so that when you begin spinning, the end of the fiber will be twisted into the yarn. This will keep your newly spun yarn from separating from the leader.