Because our place in Seattle is rather small, I have given a lot of thought as to what equipment is actually necessary, and what is just "nice to have". Going vegan has also changed what is important to some degree. So I thought I would talk a little about what have been essentials for me. As always, your mileage may vary.
Probably number one is a really good knife. It took me many years to appreciate how important having just one really good knife is, and after I got one I understood. I bought one on the recommendation of Kenji Lopez-Alt, a Misen, which I see is a lot cheaper now than what I paid. They recommend, and I agree, that you only really need three knives. A chef's knife like the Misen or a similar quality, a fairly cheap paring knife, and a good bread knife. Those three will cover 90% of the needs, especially if you are vegan. I usually go to The Wire Cutter to get good reviews of, well, everything. I don't always get what they recommend, but their write-ups usually educate me so that I feel good about whatever I do choose.
To go with the knife, a good, solid cutting board is required. I have both wood and plastic - I used to reserve the plastic for cutting meat or fish since it is easier to wash, and that would help prevent bacterial contamination. Now I just use the wood, except for chopping up chipotle peppers.
In Seattle I got a really nice deal on a set of pots when we bought a new washer/dryer at Best Buy. I got a set of Cuisinart tri-ply stainless - 2 2-quart sauce pans, stockpot and a saute pan. I would always recommend tri-ply or equivalent technology - it will help the pan heat much more evenly. I also have an iron skillet, three stainless skillets in various sizes, and a teflon coated skillet. Cooking for two, this is entirely sufficient. If I were cooking for more than two, 2-quart sauce pans would not be big enough. And in Houston, I do love my Le Creuset, but that is an indulgence.
I have just three kitchen appliances I would consider essential. First, a really good blender. I have a Blendtec, and I am very happy with it. It was not cheap, but I use it easily once or twice a week, sometimes more often, and the results are always perfect. I mostly use it for soups and sauces (dried chilis, pepitos, onion, and garlic for tacos say, or hummus, or cauliflower alfredo), anything that needs to be beaten into submission. Serious eats has a great review of high-end blenders that is well worth reading.
My second key appliance is my Instant Pot. I don't use it for most of the available functions (like cooking rice), 90% of my usage is for cooking dried beans, and for that it is perfect. It is also useful for making vegetable broth, and there are a couple of other useful recipes (risotto) but really mostly beans. Cooking for two I found it helpful to buy the small one as it is easier to work with for smaller quantities. I probably use it 4-5 times a week.
Third on my necessities is my stand mixer, a Kitchen Aid. It is really only needed if you make bread, but for that it is invaluable. I have made bread without it, but that is a lot of work and very tiring.
I also have a Cuisinart, which is useful at times, but does not get nearly the usage the other tools do. And an immersion blender is really handy, but again, I don't use it that often.
I do use my salad spinner fairly often - usually to dry spinach, or dewater cooked eggplant.
I have a full sous vide setup, which I used to use all the time. But mostly I used it for meat. It is great for cooking beef, pork, or lamb perfectly. Nowadays I primarily use it for asparagus, which it does perfectly, and sometimes carrots. I need to find some more uses for it. I have an Anova circulator, along with a tub and lid, and some weights to keep the bag submerged. Anova also recently came out with a really nice, compact vacuum sealer that I use.
For making tortillas, I have a cast iron tortilla press, made in Colombia. I think that cast iron is the only way to go, I doubt aluminum would hold up. I set up a price watch on camelcamelcamel.com until it went on sale.
I also have a nice wok, which it turns out is also a great pot for deep-fat frying. Kenji has a whole column on buying woks which is quite helpful. He is also about to come out with a new cookbook devoted to wok cooking - I'm anxious to see that one.
One thing I have started doing when cooking, is to do most of the prep (chopping, cutting, cleaning) up front, and then focus on the cooking. I burnt too many things while turned away from the stove chopping something. I bought a bunch of nice sized plastic bowls that I use for the chopped onions, kale, potatoes, whatever, while I am doing prep. I also recently bought some tiny 4 tablespoon measuring cups to use for gathering herbs and spices together ahead of time. They are working out quite well.
I should mention that I have learned a huge amount by reading columns on Serious Eats. They have a whole series on techniques and tips which has been really valuable. That's where I learned to use a kitchen knife, how to chop up an onion, dismember a pineapple, dice a butternut squash, and much more. I commend them to you. They improved my skills significantly.
I'm not a big fan of specialized kitchen tools - I think they are usually a waste of money and space, but some tools are essential, and some may be specialized but make life so much easier that they may be worth it. A tortilla press probably falls into that category.
Anyway, this is how I have outfitted my kitchen. Very dependent on the kinds of food I make, in many ways. After all, Mandarin Monday and Taco Tuesday require a wok and a tortilla press.