To start you need fiber and a drop spindle. Some people recommend to start with undyed wool that is carded, not combed. The terms "carded" and "combed" are used to describe the two major types of fiber preparation. If carded fiber is unavailable, then choose commercial combed top, preferably undyed. Dyeing can compact fibers, which can make drafting more difficult. The following sheep breeds may be an excellent wool choice for a beginner: Coopworth, Romney, Corriedale, and Bluefaced Leicester.
All these are good suggestions, it doesn't mean, however, you can't do otherwise. You should like your fiber choice, that's important. My first wool was dyed combed top, breed was not specified. It was a well-prepared fiber, easy and fun to spin. If you choose dyed fiber, it may be a good idea to avoid variegated colors at first and stick to solid and semi-solid colors (one color with shades ranging throughout the fiber). Variegated colors may require a special approach and be distracting in the beginning, so you may want to save playing with colors for later, when you are more adept at spinning technique.
Wool is the best fiber choice for beginners, since it is easier to control and spin, it's not slippery and has a long staple. Start with at least 4oz of fiber. Etsy is an excellent source for buying wool. Type in "spinning wool" in the search box and you'll get plenty to choose from.
When it comes to choosing a spindle, different people prefer different types of spindles to learn and teach. I started with a top-whorl spindle. In contrary to a low-whorl spindle, it allows you to easily stop and hold your spindle with your knees when you learn to spin and employ the park and draft method. It also allows you to roll the spindle shaft on your thigh to set the spindle in motion, which is a more powerful and faster way to add twist to your fiber, than if you flick the shaft with your fingers. It'll all become more apparent as you go along.
Spin singles first, then ply two singles, that is spin two spun threads together only in the opposite direction, to get a two-ply yarn, which after washing can be used in a knitted or crocheted project.
Here are some helpful links from Deirdre to get you started.
This website has instructions on Andean plying, but tons of other useful information and products.
This website has videos on spinning.
Useful information on spinning with a drop spindle, including Andean plying.
This web store sells a "Colonial" wool roving that I think is wonderful for learning to spin. Look under the "Yarn & Fiber" tab for the "wool roving." Their alpaca roving is also easy to spin.
This is a terrific place to buy hand dyed roving once you are ready for something more interesting.
Look for her August, 2006, entry on using a book to help you Andean ply.
Also remember that your local library can be a good resource for books on spinning.