If you want to buy the perfect pair of sticks that you'll enjoy for years to come (or at least until you have too many core shots), then the first step is to rent. Stick to renting at first. Ski shops usually offer deals for the longer number of days you rent with them, and will also usually allow you to trade in for different demo rentals each day.
Save yourself some money by asking if you can put some of the amount you paid for renting towards the purchase of a brand new pair of skis (most shops have a policy like this). Ask if they have a rent-to-buy policy. Always try before you buy.
Make sure to look at different shops that carry different brands from each other.
There are different skis for different skiers. Everybody has their own style and prefers to ski different types of terrain. For every type of skier and terrain type there is a different type of skis.
Pro-tip: If it's a local shop, tipping goes a long way and can help get you hooked up with better gear.
When looking at skis, it can be an information overload. Each brand hypes up a different feature. The key things to pay attention to are width, length, stiffness, and curve.
A ski usually lists it's dimensions in a three number set (x-x-x, tip - underfoot - tail ). The middle one is most important to focus on at first. This number is the width of the ski under your foot. If you like groomers and hard-packed snow, look toward the narrower end of the spectrum. Narrow width allows you to cut and carve into those popular slopes that can be icy in the mornings and evenings. Skiers looking for deep powder will look toward the wider side, because the increased surface area makes it easier to float on top of the snow. Somebody that want decent performance on all slopes will want something in the middle.
The other two numbers in the dimensions indicate tip length and tail length. The difference between these numbers and the width under foot will decide how quickly the skis will respond to turning. As the difference increases, you'll your movements to turn can be more subtle. The only way to know what works for you is to try it.
The rule of thumb at the time of writing this is that you want your ski, standing on it's tail, to be between your chin and the top of your forehead. The closer the ski is to your chin, the more responsive it will be, but you will sacrifice speed. The closer the ski to your forehead (or even longer in some skiers' preferences) the faster you can go, but it will be harder to control. Beginners usually aim towards chin heights, and the advanced aim towards their foreheads.
Another indicator of responsiveness, but also dependent on terrain type. Stiffer equals more responsive, but makes for a rougher time in areas with moguls and uneven terrain. It's hard to judge stiffness by feel, but the materials the ski is made of can be a good indicator. Softer woods and less metal in the core make for a more flexible ski.
The curve of the ski can be seen when viewed in profile or when it rests on the ground. If you like to ski powder, you'll want the tip and tail to be curved up more ("rocker" is the term). If you are strictly sticking to hard snow you'll want a ski that's shaped more like a mound (less curved up at the tip and tail, and more of a curved under the foot so the middle of the ski doesn't touch the ground when it rests on the floor). An all-mountain skier as usual would want something in between.
This is just the basics to choosing a ski that's right for you. There are more factors that contribute to the feel of the ski (such as tip and tail shape for one), but the only way to really understand the differences these factors make is to get out there and try as many skis as possible. Do your research before any purchase and ask for advice, but be aware that a different opinion comes with every skier.
Ski safe and have fun!