It's tempting to purchase a full set of shiny new pots and pans. But your choices should be determined by how you cook, not what a maker can fit into a package.
Copper pots and pans are the ideal: They channel heat perfectly, last forever, and look unbelievable. But they're prohibitively costly and difficult to maintain; most people shouldn't even consider them.
There are other good choices. Cast iron conducts heat nearly as well at a fraction of the cost and is pretty much nonstick to boot. With the increasing worries about nonstick coatings, I now cook almost everything in well-seasoned cast iron. But cast iron is heavy (make sure large pots and pans have handles on both sides), and the iron itself can react with some foods and discolor them. (Enamel-coated cast iron takes care of this problem, though at a cost.)
Long-lasting and fine-looking, stainless steel is a "neutral" metal, meaning you can cook anything in it without worrying about pitting or reacting. (This is what is meant by the term nonreactive.) It's great for stockpots and saucepans, where you're working with mostly liquids and where cast iron can cause discoloration or even adversely affect flavor. Saut? pans and skillets are fine too, as long as you accept the fact that food will stick to their surfaces unless you use at least a thin coating of butter or oil and correctly heat the pan.
Stainless should be high quality; the cheap stuff is usually too thin to conduct heat properly. The best have bottoms made by wrapping relatively thick stainless steel around a core of aluminum and/or copper; the combination conducts heat evenly, prevents warping, and minimizes burning. These pans should be fairly heavy. If not, keep looking.
Aluminum is another popular material for cookware, provided it has been anodized, a process that hardens the metal and makes it more durable and less reactive. Even though various cookware lines might look similar, quality and prices can range wildly, so be sure the metal is thick, especially on the bottom.
Ceramic cookware is fine for oven-braising, gratins, and baking, but with rare exceptions, you can't use it on the stove. Don't even bother with glass pots and pans. They shatter when you least expect it, and they're worthless for anything but boiling water.
A word about nonstick coatings: Like most people, I've become convinced that concerns about their safety at high temperatures are warranted. So I don't buy them anymore; cast iron, believe it or not, really does just about as good a job of preventing sticking.
Handles and Lids
Good pots and pans have their handles attached by rivets, and those handles are made of metal; wood and plastic are functional enough as long as you know the pan will never ever go in the oven or broiler. But how would you know that? Stick with metal handles.
The more pans you have with lids, the better. And though the material isn't that important, once again metal is best.