Sharp Stuff

A chef's knife. This is your most important piece of equipment in the kitchen. Get the best you can afford and take care of it. I use a Bob Kramer 10" Chef's Knife because I'm overcompensating. I've found Victorinox knives to be very good and very cheap.

A good paring or utility knife. For cutting small things, hulling strawberries, etc.

Cookware

A large skillet. You don’t need fancy stuff. Cast iron is fine. Clean it with soap and water like you would any other pan, and never store it wet or it’ll rust. I dry mine on the stove over high heat. Use a paper towel to rub it with oil when you're finished and it'll almost become nonstick.

A small nonstick skillet, for eggs.

Utensils

A wooden spatula. The flat surface of a wooden spatula is far superior to a wooden spoon for deglazing a pan.

A silicone spatula. I love the bendiness of silicone spatulas. Find one that is sturdy but comes to a sharp edge. This will allow you to mix things better and clean out mixing bowls. (Although, if you ask me, I never minded leaving excess batter in the bowl as a little treat for myself.)

Tongs. For removing larger, studier items from a pan, and transferring long noodles from the pasta water to the saucepan.

A whisk. To whisk things.

A spider. Use these to remove things from cooking liquids, like pasta, dumplings, etc. Your days of lifting and moving a huge pot of scalding pasta water over to the sink, crossing your fingers while you pour, and then remembering you were supposed to reserve some of the cooking liquid are over.

Measurement

A digital scale. Get one that measures in grams and ounces.

Bonus points if you also pick up a jewelry scale for weighing small amounts, like salt, spices, or cocaine. You'll never have to use measuring cups again.

Instant thermometer. Get a Thermapen. It’s the last instant thermometer you’ll ever buy.
Probe thermometer. This takes the guesswork out of roasts.

Prep

Graters. I love Microplane. I recommend their Professional series without all the plastic parts because the plastic parts break. 

Mixing/prep bowls. I use stainless steel one-cup-sized prep bowls for prepping. All my diced produce goes into these as part of my mise en place.

Tamis. Two reasons to get a tamis: Mashed. Potatoes. It’s a tambourine-shaped fine-mesh sieve that'll transform your potatoes [and your waistline]. You'll also need a silicone bench scraper to push the potatoes through. 

Fine-mesh sieves. I have three: a small one to double-strain cocktails, a medium one for sauces, and a large one for stocks.