1. Tongs: for tossing salads, not burning the crap out of your hands flipping hot stuff, pulling pasta out of hot water, endless uses. These are indispenable. How did I ever live without them, I think to myself every time I use them. That good, I swear.
2. Magnetic measuring cups and spoons. Magnetic ones don't have that goofy ring that you have to remove to avoid getting the whole set dirty when you use just one; they stack neatly in a drawer; and besides, I'm a sucker for all things magnetic (e.g., knife storage-- not essential, but keeps knives within easy reach while avoiding dulling and potentially hazardous drawer storage).
3. Set of stacking mixing bowls. Glass, plastic, metal- material is a matter of personal preference. Actually, two sets is helpful if you find yourself doing a lot of cooking. (This is true of the tongs and measuring cups/spoons as well-- nothing worse than having to wash something mid-recipe.) I just bought 2 big cheap-o metal bowls and have used the hell out of them, proving that stuff doesn't need to be expensive to be useful. (Just look at the stuff that professional restaurant kitchens use.) I will contradict myself shortly with an admonition about buying good stuff, but it really depends on the item in question.
4. Knives. A large chef's or Santoku knife, paring knife and bread knife (great for tomatoes and other soft-fleshed fruit/vegs) are all you need. Buy good ones and they'll last a lifetime. (Mine were a gift to my now-husband before we were married: a sign of confidence and investment in the relationship. It's worked out well.) Get a sharpener, too, while you're at it.
5. Stock pot or Dutch oven. Soups, stews, braises, boiling water for pasta. Go for maximum volume and sturdiness. Dutch ovens are freakin' heavy, and if that will deter you from using it, buy a cheap aluminum pot-- it boils water just as well and you can always step up to something swankier later.
6. Sauce pan. Good for boiling eggs, heating up a can of soup, or more ambitious projects.
7. Cast iron or enamel frying pan(s). I have a few-- different sizes and materials. My cast iron skillet was my grandmother's-- a gift from my grandfather early in their long and happy marriage, I believe. They lived a simple but rich life together-- the pan has a humble functionality that reflects that life. My two smaller Le Creuset pans are a kooky orange and were a wedding gift to my mom when she married my dad. The pans outlived the marriage by a long shot, look somewhat dated (some shades of orange can't help but shout "70s" at you) and are permanently stained from use (mine, not my mom's-- she never ever used them-- too fancy, heavy and impractical for her). All three of these pans are in heavy rotation in my kitchen and illustrate the importance of buying good stuff-- you'll be saving your kids money in the long run. Unless a) your kids are brats who insist on having the latest gadget, in which case they don't deserve your "heirloom" pots and pans, or b) you don't have kids, in which case you'll make some lucky bastard at the Goodwill insanely happy when he finds your culinary treasures in the bargain bin.
8. Pyrex baking dish (~9x13). For lasagna, cakes, marinating, etc. Love that you can see through the bottom to gauge how quickly things are browning (or sometimes in my case, burning). It's like a super-power of cooking.
9. Vegetable peeler. You can use your paring knife, but why be a martyr?
10. Instant-read thermometer. Essential to determining "doneness" of meat until you learn how to judge by touch. If you're a vegetarian, I've recently learned that an instant-read thermometer is one of the best ways to tell when bread is fully baked. I haven't found a great one that is truly instant read, making for slightly more guess work than I am comfortable with re: internal meat temp (this coming from someone who likes medium-rare hamburgers and steak tartare in restaurants, but is not likely to feed same to her family or self at home).
11. Strainer. For draining pasta, washing vegetables, wearing on your head like a hat (if you're almost 3 years old like lil' P.).
12. Mandoline. Not strictly necessary, but if you find yourself doing a lot of fine slicing, especially of onions, this will make quick work of something that otherwise is a chore. Just watch your fingers and save yourself a trip to the emergency room.