The equipment needed for dry slope racing splits into two categories, namely skiing equipment and safety equipment. Like all sports, you can spend an almost unlimited amount of money on equipment, but fortunately you don't need to as most of the equipment needed can be purchased relatively cheaply, and there is often good quality second hand equipment available. This page gives a brief description but all the details are in this document. This is what you will need;
Don't be put off by the amount of equipment needed. Dry slope skiing can be as expensive or low cost as you want it to be. Don't feel you have to buy all the equipment at once and don't feel that second hand equipment is somehow inferior. It is possible to get kitted out over time for a very reasonable amount of money. And please ask for advice, all the Vikings - racers and coaches - will be very happy to advise you and help you find the right equipment within your budget.