There's about five tools that I use to cook just about everything. They are:
8" chef's knife. It's worth spending $100 on this, since they last a long while and get daily use.
I've had my J.A. Henckels Zwillingswerk knife for . . . gosh, 14 years? I love it. Wustof and Global are also good choices. I've used a 6-inch Henckel's chef's knife for a while at my parent's house and might even prefer it to the eight inch; when I get another knife that'll be the one. A lot of people will tell you that a paring knife is important to have. I have one, but I only use it when the chef's knife needs to be cleaned and I'm lazy.
Twelve inch stainless steel skillet or sauté pan. It should be silver, because that's the color that stainless steel is. Not black or nonstick. I have an All-Clad; if I was on a budget I'd get the Sur La Table house brand.
If properly used, stainless is even more non-sticky than nonstick, and personally I'd like to minimize the amount of polytetrafluoroethylene in my life and my bloodstream. The way to clean a stainless steel pan, and possibly produce a basic sauce in the process called "deglazing": immediately after you have emptied it of food, and turn off the heat, add a few ounces of beer, wine, or water to it. You'll get a hiss. Then scrape the bottom with your spatula. Oftentimes this is a delicious stuff to add to whatever your making. At the very least, it helps clean the pan. Then, unlike nonstick, you can use mechanical abrasives like Bon Ami and/or Chore Boy copper scrubs to get the rest off.
The difference between a skillet and a sauté pan is that the former has curved sides, while the latter has vertical sides. The sauté pan has more volume; the skillet allows you to flip things more easily and is slighly lighter. Either is fine.
You'll also want a lid. Preferably transparent so you can see what's going on inside. Either buy a pan that comes with one, or I got a translucent silicone lid that's been great.
Two Le Creuset pots. I got the Creuset "essential set" with a 3.5 quart pot that is super useful for cooking grains; a 1.5 quart saucepot that is also useful for cooking grains and vegetables although I often wish it was larger; and a 9 inch skillet that makes the most perfectly browned pancakes and omelettes because it has such good heat distribution. Really the pots are the most useful, but the set's cheaper and the skillet has been useful.
A pot to boil water in. Stainless steel again, but often less expensive because the bottom doesn't have to be as thick as with a skillet, since it'll be full of water most of the time that will distribute the heat instead.
Tongs. Cost about $5-10. Excellent for turning things over or grabbing stuff out of boiling water.
That's it. And a cutting board.