Here's a list of kitchen things, sorted in sections and (my view of) priority order. Though what would be useful for you is somewhat personal, depending on what you tend to cook, I hope this can be helpful for you! Let me know if you have any questions/comments/concerns/clarifications.
COOKWARE: Heavy-bottomed, oven-safe stockpot Why? To make your own broth, cook anything you want in bulk, frying (if you do that), or literally just put a chicken in there with water, boil, and leave it in the oven for a while and you've got broth and chicken dinner! Also heavy-bottomed is more expensive but it gives more even heat and will last WAY longer. Seriously. I have a thirty-year-old heavy-bottomed stockpot that's like sparkling new. (alternately, I would suggest a dutch oven, but good-quality ones are a ~500$ investment and a solid stockpot with a lid will cover most of what it does, albeit less beautifully) Large, high-sided, nonstick frying pan Why? Nonstick because you don't want to make cleaning any harder for yourself (you've got lots of studying to do!), huge because you don't want to be crowding food on your pan, and a bigger pan is usually only slightly more expensive than smaller ones, and it's terrible to try and fit too much on a tiny pan. High-sided so it can double as a sauté/sauce pan. Also, for pancakes and bacon so you aren't splattered with grease nonstop. 9x13 Casserole/Roasting Pan (With a lid!) Why? Maybe the least essential of this list, depending on your cooking habits, but it's been very useful in my experience. Use it to make bread, cook a lot of meat at once, make a casserole with all that chicken from your stockpot, bake brownies, reheat leftovers, and most importantly, store and freeze leftovers. Make sure you get one with a well-fitting lid (preferably silicone rims). Now, onto:
STORAGE A bunch of thick glass containers. That's it. Why? Glass can go from an oven to a freezer to a microwave with no issues, and makes it easy to refrigerate/freeze smaller portions. Good quality (I recommend Pyrex or Anchor) also lasts a long time and won't shatter when you drop them. A few 2-cup, 4-cup, and one big 2L (in that order of priority) should be good. Disposable Storage: Freezer bags Why? Store mini meals or excess produce (stir-frys are a great option) to be thrown onto your nonstick skillet and cooked quickly. Although they're disposable, they're also pretty indispensable for dealing with overflow. Aluminum foil Why? Very all-purpose, for wrapping things up, baking, making impromptu lids, baking, heating things in an oven, baking (seriously though, make your own muffin tins, grease&flour it to replace parchment paper, cook many things in one container and use this to separate them: very, VERY handy for anything with ovens.) Alcan is a strong brand. Plastic wrap Why? Sometimes your food is just a little too tall and the lid won't squish it down. Not really necessary when you have freezer bags, but good to have on hand.
PREP: Hard Plastic Cutting Boards Why? Don't ruin your plates (or worse, your countertop)! Your knives will leave marks. That's so the boards don't dull your knives. Still fine to use. Cheap ones from IKEA that you can throw into the dishwasher work fine. I like to have two or three, one for raw meat and one for cooked food/fruits and vegetables/baking. Do what suits your needs best. DON'T use wooden or marble slabs unless you're prepared to maintain them constantly. Glass Measuring Cup Why? Yes, you could buy that cute stack of nesting measuring cups, and go for it if you want that precision, but a 4-cup graduated glass pitcher with clear markings will probably do you just fine. Metal Measuring Spoons Why? Eyeballing things will only get you so far. If you bake or want precision in small quantities, these are the way to go. Not necessary, but nice to have! Fine Mesh Sieve OR Colander Why? To drain water out of things. If you make a lot of pasta or wash large quantities of fruits/veggies, go for the colander. If you need to aerate flour or make perfect icing or strain cheese, go for the fine mesh sieve, which does everything the colander does and more, except doesn't have feet to stand on. Again, not necessary, but useful if your kitchen habits and budget desire it. Mixing Bowls Why? I'm actually /excluding/ these because you'll have rounded glass containers of suitable sizes. But, if you bake or make huge salads regularly, consider including a mix of small and large non-reactive bowls with lids (or cover with plastic wrap). By no means necessary, though, and usually even cereal bowls work in a pinch.
UTENSILS: Knives: Three good knives should serve you well for basically all your kitchen tasks. Chef's/Utility Knife Why? Good for cutting most meats and everything with a hard skin. Get one with a nice grip that feels good in your hand. Serrated/Steak Knife Why? This cuts everything that the chef's knife can't. Just try cutting a tomato with a sharp-edged knife. The teeth get through soft skin and make your life so much easier. Also: cuts steak. Paring Knife Why? With a sharp, short, and thin blade, this is good for smaller goods, where you don't want to wield something so huge. Also handy to cut fruit for a snack or disengage something from the side of a cooking vessel. Not really essential, but handy to have around. Other Utensils: Silicone Spatula Why? Flip, push, fold, whatever manipulation you have to do in any cooking, a good silicone spatula should do the trick. Try for a wide one with thin edges to get under food. A flexible fish spatula works too, but not so good for pancakes, sauces, or anything drippy. Hard Spatula Yes, again. Why? A firm one with sharp and curved edges works better than a wooden spoon for stirring thick mixtures and helps you get the last drops or fond from anything. Not as necessary as a thinner silicone spatula, but it's good to have. Tongs Why? Lift and flip large things easily, and toss salads. Get sturdy ones, or they'll be useless. You could use two pieces of cutlery, but that depends how close you're comfortable getting your hands and arms to your hot food-to-be. Balloon Whisk Why? Some argue that a fork does the job just as well, but if you want light and fluffy eggs or cakes, a smooth sauce or curry, or anything to be aerated and well-mixed, a whisk is essential. If you don't care, that's fine too though! Another alternative is a ball whisk, but they tend to be somewhat more expensive. Check with your nonstick pan to see if it needs silicone utensils or if metal ones are fine. Cutlery: Knives Why? Covered above. I don't think you'd need dinner knives too, but many people want them, so I'm including them here anyways. Forks Why? To stab things with and use as a whisk in a pinch. If you're not getting a whisk, make sure your fork has long tines (the pointy things). Spoon Why? To slurp things. Cereal and soup, basically.
EATING: Plates, Bowls, Cups. Why? To eat from!
ACCESSORIES: Oven Mitts Why? So you don't burn yourself! Fitting your hands and non-slip grips are pluses. This is essential. Don't think towels will do the job. Apron Why? So you don't splatter yourself! Not essential but useful if it has a pocket or you cook in clothes you care about or ever fry things or cook bacon without water.
CONSUMABLES: Paper Towel Why? Clean up spills and messes and grease and wipe things clean. Dish Soap Why? Clean all your food stuff. I love Palmolive, but any brand is probably fine. If you have a dishwasher, don't forget dishwasher soap, which is different. NEVER put dish soap into a dishwasher. Youtube what happens if you don't believe me. Baking Soda and White Vinegar Why? Multipurpose everything-cleaner. Doing a quick search for baking soda cleaning should give you quite a few results.
it was too long, so part 2 is upcoming!