Before you move into your new apartment, while your parents are dealing with their empty nest syndrome and are still willing to buy you things, make sure you realize how many things you take for granted in your cabinets. Apartments don't just come with ketchup and frying pans (unfortunately). Here is my list of things you should get while you still have cash.
-Pots and pans (one big sauce pot, a medium sized saucepan, and a skillet are most important; a small skillet would also be good if you plan on cooking omelets!)
-Pie dish (even if you don't plan on baking pies)
-A couple sharp knives
-Cookie sheet (or something that looks relatively like one, it's helpful it you get one with sides so you can cook other things in it)
-Baking soda (always have this on hand even if you don't cook with it because it is the way you stop grease fires. DO NOT use water.)
-Vinegar (I use apple cider vinegar)
-Spices (my most frequently used are: salt, pepper, garlic powder, red pepper flakes, chili powder, Italian seasoning, and oregano)
-Garlic (it's way more cost effective to just get a huge jar of pre-minced garlic, especially if you love garlic as much as I do)
-Your favorite condiments
-A couple of your favorite salad dressings
-White sugar and brown sugar
I'd recommend trying a Building 19, Ocean State Job Lot or a dollar store for pretty much everything when apartment shopping. Yes your hands may feel dusty and gross after going through all of that stuff, but you can get reasonable things for great prices.
I was also browsing Amazon and found this which looks to be super cheap and has a lot of the things you'll need. Farberware Clear Benefits 14-Piece Cookware Set, Black