Style
The most important decision you'll make when purchasing a new pair of skis is what style you'd like to have. To answer this question, consider what kind of terrain you most enjoy skiing. Most skiers get a versatile all-mountain ski, but more experienced skiers may look for powder skis, freeride skis or telemark skis.
Powder skis are great in deep powder, like the champagne snow of the Rocky Mountains or the deluges of the Pacific Northwest. Freeride skis are best for those who spend their time in the terrain park or half-pipe, but they also transition well on to the rest of the mountain. Telemark skis are becoming more popular for in-bounds skiing, but are really meant for the backcountry.
Length
The length of the ski you'll need is mostly determined by height and ability. The taller the person, the longer the ski. However, this is offset by ability; the more novice the skier, the shorter the ski. There's no hard-and-fast rule that applies lengths to skiers, which is why it's important to visit a ski shop. Shorter skis are easier to turn and to control, whereas longer skis are better for those who like to move fast, but have a good command of their skis in irregular terrain. As a general guideline, skis should be about the same length as you are tall or a few inches shorter.
Boots and Bindings
Choosing a pair of skis is a great place to start, but you also need to select quality boots and bindings before you're ready to hit the slopes. Bindings keep your boots attached to your skis and are designed to release the boots in any jarring and potentially damaging situation, thus keeping your leg and all its ligaments intact. The most important consideration for your bindings is the level of aggression you put in to your skiing. Newer and less aggressive skiers can go with a relatively inexpensive and lightweight binding, whereas more advanced skiers will want to get a lightweight, durable binding that offers a higher release setting.
Boots are arguably the most important part of making a ski package purchase. They should fit snugly but shouldn't have pressure points anywhere in the foot or lower leg. Beginner boots are softer and have a lot of flex, allowing the skier to comfortably control the ski. As a skier increases her ability, she should also increase the stiffness of the boot. Increased stiffness means better response from the system and more precision in skiing.