Sharp Knives. Everyone has their own preference on style and manufacturer, so pick the ones that work for you and keep them sharp. I rarely use a chef’s knife, I use a paring knife for small jobs and a cleaver for just about everything else, except sushi – I do have a sushi knife and it doubles as a carving knife.
Cutting Boards. I think we have 7 cutting boards – 2 huge ones, 3 large and 2 small – we use them all. The smaller ones have a hole to hang them so they are always handy hanging above the counter they protect.
Salad Spinner. You can live without one, but it does make prepping greens much easier.
Colander. I hate mine – it has a small lip inside that catches things when I pour them back out. My advice would be get one without that. For cold items I use the inner part of my salad spinner.
Sieve/strainer. I like the 1½ cup size for most things but also have a ½ cup one that is great for herbal teas. A stainless steel mesh won’t stain like nylon can.
Grater. Even if you have a food processor, a good hand grater is still useful.
Pots and pans. A skillet, a medium-sized pan and a large pot would be about the minimum. A steamer basket is useful, but if you have a metal colander you could use that instead.
Roasting pans and baking sheets. One of each would be good for a minimalist kitchen.
Storage containers. Whether you use glass or plastic, jars or bags, you will want something to store prepped foods and leftovers in. I love my Pyrex and my canning jars, but other people swear by the deli type plastic tubs and Ziplock bags.
Measuring cups and spoons. Not really vegetable related, but if you follow recipes you will need to measure your ingredients.