These are the basic tools needed to be successful in the kitchen. Your suggestions are welcome!
  1. A sharp chef's knife
    A sharp, good chef's knife is really all you need as far as knives go. It should be comfortable to hold, and you should make sure to have it sharpened at least once a year. if you're willing to spend the money, a few good brands are Wusthof, Shun, Vitorinox, Masamoto, and Global. But a cheaper, non-fancy, $20 one that you keep sharp will do fine.
  2. 2-3 plastic cutting boards
    Wooden cutting boards are great, but when you're first starting out, plastic ones just make more sense. I especially like the lightweight, flexible ones that can be lifted and bent to pour ingredients directly into a bowl or pot.
  3. Salad Spinner
    For drying lettuces but also to get the right amount of oil, vinegar, dressing on the leaves when making a toss.
    Suggested by @michelle
  4. Kitchen scissors
    These are great for cutting up bunches of scallions, snipping fresh herbs, cutting up a chicken, and even slicing up a fresh pizza.
  5. A cast iron pan
    Buy a 9" or 12" cast-iron pan with high sides, and don't bother spending a lot of money. If it's cast-iron, it's going to last forever. Army & Navy stores have a good deals on these. I store mine with a light coating of oil on it, and avoid washing it with soap. When it's dirty, just rinse it and wipe it down.
  6. 2 nonstick pans
    These are great for scrambles, omelettes, stirfries, sautéing, etc. Get a smaller one for omelets (6") and a bigger one for larger dishes (12").
  7. 2 pots
    One pot that's big enough to boil pasta or make a big pot of soup, and a small pot for boiling eggs, making oatmeal, and cooking gravy.
  8. A Dutch oven
    An ovenproof large pot with a fitted lead like the ones available from Le Creuset or Dansk are excellent for making chili, stews, and long cooking sauces. You can use this just like a slow cooker if you keep it over a low flame.
  9. Things to stir with
    Three or four wooden spoons, varying shapes and sizes. A rubber scraper spatula, a sturdy whisk.
  10. Basic gadgets
    A vegetable peeler, a can opener, measuring spoons, measuring cups. I like to have one Pyrex liquid measuring cup around, and then a set of dry scoop cups for measuring things like flour and rice.
  11. A meat thermometer
    Once you been cooking for a while, you can usually tell when it is done by looking at it and touching it. However, if you like to be precise, a meat thermometer is a good idea. It does not have to be fancy, and should not cost more than five or six bucks.
  12. Tongs
    Buy one pair of tongs with silicone tips, and one with metal tips. These are amazing for everything from fetching chicken off the grill, to tossing a big salad, to transferring pasta from boiling water into a simmering sauce.
  13. A rolling pin
    I use my rolling pin for everything from the obvious (rolling dough) to the less obvious (crushing toasted nuts in a ziplock bag to tenderizing meat). That said, there have been plenty of times when the rolling pin hasn't been within reach, and so I have used a clean wine bottle instead.
  14. Bowls
    A set of mixing bowls that includes a few very small bowls for things like sauces and chopped onions, garlic, and herbs will come in handy an endless number of times in the kitchen.
  15. Bonus: immersion blender
    I cannot say enough good things about my immersion blender. First of all, it's several appliances in one. I use it to whip cream, make soups and purée sauces, directly in the pot, blend smoothies, chop onions and garlic, make pesto, and so, so much more. Even better, it's about the same size as a rolling pin, and can be stored in a drawer. I got mine for about $40 at Bed Bath & Beyond (and it would have been cheaper if I had remembered that BBB coupon!).