This post will not attempt to discuss what you will need if your dorm room (or other habitation) lacks a kitchen. Vanessa is much more prepared to discuss the details of hot-plate cooking, talking your way into other people's freezers, and the care & keeping of mini-fridges than I will ever be. This is simply an account of what tools you will need to cook good food and a few fancy treats.
Stirring, Mixing, and Beating:
A few good spoons. I prefer wood and plastic spoons (metal scrapes too much).
A medium-sized wire whisk. Good for everything from scrambled eggs, to custard, to hot chocolate. I use one to combine dry ingredients for baking.
A rubber spatula. Or two, or three. Don't buy the expensive ones formed for scooping. You want it flat and flexible for scraping sauces and batters off of bowls and pans.
A pasta fork. You know, those slotted things with the teeth. (Never strain out the pasta water, unless you've made really small pasta. Fish it out with your pasta fork. The little bit of pasta water adds body and flavor to any sauces and keeps your pasta from sticking together.)
A spatula. Not strictly for stirring, mixing, or beating, but useful to have around for cookie baking, fried eggs, and most anything else.
Metal tongs, for removing items from boiling water, frying oil, or the barbecue.
A ladle, for serving soup (or Wassail).
Electric Beaters. Share them with your roommate, a friend, or the girls across the hall. You won't always need them, but when you do, you'll be happy to have them.
Cutting (The Basics):
Two cutting boards. One plastic that you can clean really well for cutting raw meat and particularly odoriferous vegetables. One wooden for bread, cheese, fruit...it should be pretty enough that it can do double-duty as a serving board for bread or cheese at the table.
One well-balanced knife that feels good in your hand. The 10'' is the basic chef's knife, but I have small hands and find that I'm uncomfortable with anything much longer than 8''. If it's not comfortable, it won't be safe. Keep it sharp. You're more likely to cut yourself with a dull knife.
One good pair of kitchen shears, useful for opening packages, cutting up herbs, and cutting up poultry, among other things.
A paring knife, good for any small jobs your chef's knife can't handle.
A vegetable peeler--again, choose one that feels really secure & comfortable in your hand.
A cheese grater with holes of various sizes.
Cutting (Fancier):
If you are going to be baking (or spending good money on high-quality) bread, I really, really, want you to invest in a serrated bread knife. It will preserve the shape and crumb of your loaves and make your life easier.
A Cheese plane. If you cook & eat much hard or somewhat-hard cheese (anything from Jack to Parmesan), this helps you make nice slices for sandwiches, pizzas. It's spatula-like shape works well for serving slices of pie, cake, or pizza.
You do not need a pizza cutter.
Bowls and Bowl-shapes:
A Set of Mixing Bowls of various sizes. Metal is lighter & cheaper than ceramic, and chills well for making whipped cream and similar items.
A colander or strainer for washing and straining vegetables (and the occasional pot of macaroni).
A bowl-shaped sieve for baking. Also very useful when washing rice or legumes.
Measuring:
Measuring cups, dry. 1 c, 1/2c, 1/3c, 1/4 c.
Measuring cup, wet. The pyrex ones are my favorite. 1 c (4 c is nice but not necessary). Plastic is bad for measuring hot things.
Measuring spoons. 1 T, 1 tsp, 1/2 tsp, 1/3 tsp, 1/4 tsp.
Some stores now sell measuring spoons labeled "pinch", "dash", etc. This is stupid. Don't spend your money on them and make fun of people who do.
Pots and Pans:
A Soup pot of several quarts.
At least one frying pan. Avoid Teflon. It's not good for you, and chances are either you or your well-meaning roommate will scrape off the coating. I love cast-iron skillets, which are good for just about everything. They're heavy and costly, so buy a cheap pan or two and keep an eye out (several eyes: commission friends and relatives) at rummage sales. It will need to be seasoned with oil, and avoid using detergent in it, which will remove the seasoning. Use salt if heavy-duty scrubbing is necessary.
Several sauce pans of various sizes. These don't need to be particularly fancy. They just need to be with lids and without holes. If you get into making fancy sauces you'll want one with a good lining, but you don't need that now.
At least one rectangular baking pan, useful for cakes and brownies.
At least one pan for roasts and casseroles. I have found that pyrex is cheap and versatile, although you will collect a great variety of pans for various purposes over the years. Cast-iron enamel is not cheap, heavy, and even more versatile. Pyrex is the college version, cast-iron enamel is the grown-up version.
A pyrex pie plate, preferably one that comes with handles in the shape.
A stock pot, for the big things like pasta, soup stock, and corned beef.
At least one cookie sheet
Baking:
A rolling pin. I like the one-piece European style.
I entreat you, if you will be doing any baking, to invest in an oven thermometer. Shoreland ovens tend to be too hot, which will burn your pies and kill your yeast. You will save yourself much heartbreak and bad bread if you spend ten dollars on a thermometer now.
You may also want to invest in a cloth or silicone rolling mat. Depending on how much free space you have, this can be very useful for rolling out pastry and kneading bread.
Muffin tins are nice, but not necessary unless you see muffins and cupcakes in your future.
A pastry brush is nice to have around for applying glaze or butter.
Things nobody remembers:
Dish towels, hot mitts, and a kitchen timer
Trivets for hot pots and pans
A can opener, a churchkey, and a corkscrew
A good, cotton, kitchen apron. Yes, you'll look like a dork, but if, like me, you tend to wipe your floury hands on your lap while you bake, an apron can save both your pants and your dignity.
If you drink tea, get a tea pot. You'll get more and tastier tea for your money.