Buy It For Life

My collection has been picked up over about 25 years and is about 10 different companies and quality levels. If I had to rebuy pots and pans, here's what I would do for minimum coverage of most of my cooking:

• 3 frying pans, 8-10-12 inch. The 8 would be hard anodized, the 10 a cast iron, and the 12 a stainless steel.
• 2 pots: 2-quart sauce pot and a dutch oven. The sauce pot would be anodized and the dutch oven would be either cast iron or ceramic-coated cast iron
• 1 wok, carbon steel

Calphalon "Select" or "Premier" Hard Anodized

pros:

• Good non-stick coating
• Durable enough to throw in the dishwasher
• Stacking flat lids are weird until you realize you get 2x as many pots and lids in your pot cabinet
• Pretty heavy and heat evenly

cons:

• Not induction compatible

Alternate: All-Clad HA1, which is more expensive but induction compatible.

All-Clad D3 Stainless

pros:

• Basically indestructible
• Very heavy
• Induction compatible

cons:

• Not cheap
• likely will take the most effort to clean if they get stuck-on food (but see below)

Lodge cast iron

pros:

• Inexpensive for the features
• Extremely heavy
• Nonstick gets better with use and is fixable
• Induction compatible

cons:

• Don't put it in the dishwasher
• Requires the most care and feeding
• Cheaper ceramic coated dutch ovens will chip and stain

Alternate: Le Creuset ceramic dutch ovens if you want better ceramic durability and can afford them.

Basically anywhere I would be making "sticky stuff" like eggs, caramel, jam, or similar I want anodized. I am one of those cast-iron elitists but even I don't always want to take care of it afterwards, so I would get a 10" cast iron skillet for general-purpose use. the 12" stainless is great for almost everything saute/pan fry and searing, and if it gets covered with stuck-on food a spray of Dawn Powerwash (if you haven't used this stuff, I worry that it's made from radiation and small children but it is nearly magical) and a steel wool pad and you're clean again. Cast iron pans are good for sear-to-oven steaks, pan pizzas and cornbread outside of regular stovetop duty.

Saucepots often get sugary/sticky stuff in them so anodized is the way to go there. If I had to get one large pot it would be a dutch oven. I would prefer an anodized stock pot and a dutch oven, but dutch ovens are extremely versatile. I have an enamel-coated cheap one and a large "regular" cast iron one that's a family heirloom and use them for soup/baking/bread and even another pan in a pinch. If you get lids that can go in the oven they are a bread-baking machine.

Since I have an outdoor gas eye and I recommend if possible you get one if you like anything remotely Asian, I have a large carbon steel wok for stir fry type dishes. These, like the cast iron, take some care and feeding but like cast iron they just keep getting better with age. This is one time I don't get the heaviest thing, since there's so much heat involved it's less important and you're throwing the pan around a lot so lighter can be nice. I recommend one with a removable wooden handle so you can throw it in the oven at 350 wiped with oil a few times to build up a starting coat.

Everything else is bonus, really. I tend to look for the heaviest induction compatible stuff I can as I have an electric stovetop but I have one portable induction "eye" and it's so useful I will likely get an induction cooktop when I upgrade. Cast iron is not hard to take care of and can be extremely non-stick, as well as being totally safe for cooking with zero possibly dangerous chemicals, but it does take more work than just throwing it in the dishwasher. One of those little chainmail pads will usually solve the problem though as they get slippery once used, but you have to dry them after washing unlike everything else that I can drip-dry on a rack.

If I was only allowed one pot material it would be cast iron, but like I said I know I'm biased since I have a couple heirloom pans and am spoiled. Since heavy pan = more thermal mass = less temp drop when you put food in, you get better browning so whatever you get, just the overall weight compared to other pans of the same size is important.