What Every Cook Should Have

Salt (I prefer sea salt - the flavor is better. Get sea salt that looks like regular salt though, not the big chunks)
Pepper (ground black pepper. Or you can buy a cheap pepper mill from someplace like Walmart and then buy whole black peppercorns to fill it - that way you have freshly ground pepper)
Garlic powder (I prefer fresh garlic, but I always have this on hand. Almost everything's better with garlic in it!)
Italian seasoning (a blend of basic herbs)
Vanilla extract (get the real stuff, not imitation)

I usually get the five items listed above at a wholesale warehouse like Costco or BJ's or Sam's Club (although sometimes they don't carry Italian seasoning, but every grocery store does).

Olive oil
Vegetable oil (or canola oil - they are pretty much the same thing)
White vinegar (don't need a huge container, but you should have it)
Apple cider vinegar (don't need a huge container, but you should have it) 
Brown sugar
White sugar
Flour (a basic white all-purpose flour. If you're health conscious, you could go with white wheat flour or organic flour. If you have a gluten allergy in your house, corn starch can work as well in many cases)
Baking powder
Baking soda (yes, this is very different from baking powder)
Cinnamon (ground) 
Ketchup
Mustard
Soy sauce
Worcestershire sauce
Butter (don't cook with margarine. Butter has more flavor and works better in recipes)
Eggs (large or extra-large)
Mayonnaise
Parmesan cheese
Shortening
Pam (or some other type of spray oil to grease pans)

As you cook more and more, you'll add to this list, but this covers the basics.