Since "skiing" is the name of the sport, buying skis seems to be a logical first step to get involved. Ski instructors and shop sales persons will tell you however, that beginners should rent before they buy. When it comes time to buy they should first invest in ski boots and purchase their first pair of skis later in the process. Here's why:
Boots are first because they are the interface between the skier and the ski, and a custom fit is critical to both performance and enjoyment. Buy the best boots you can afford and seek assistance from the shop's technical employees to get them fitting comfortably. If the shop or outlet where you bought them does not offer capable boot-fitting, you can take boots you bought elsewhere to a professional boot fitter for custom fitting for a fee. For more information on why ski boots are so important, check out this article.
We recommend holding off on buying skis so you can audition different models for the "Goldilocks" ski that feels just right to you. Start off with very short skis--approximately chin height--and gradually increase the length as you improve to a length somewhere between your nose and forehead so the skis can handle the increased demands of faster speeds and stronger forces. For more information on choosing a ski length, check out this article.
Usually ski poles are included with skis when you rent. Poles are the least expensive piece of ski equipment, but also come in an array of models, styles, and price points with various features and benefits to appeal to the gamut of skiers, from the entry level beginner to the World Cup racer. The important thing for the beginner skier is to get them the right length. To size poles properly, up-end the pole so it is perpendicular to the ground, where the handle is pointed straight down and the tip is pointed straight up, and grasp the pole in the palm of your hand between the tip and the basket. If the angle of your arm at the elbow is approximately 90 degrees, it's the correct length for you.*
* In the past, ski instructors and shop sales persons would advise that the pole be gripped just under the basket when using the aforementioned ski pole sizing technique, because we used to initiate turns by "planting" the pole in the snow rather than "touching" the tip on the snow.