A good size, high-sided cast iron skillet and a decent sized cast iron dutch oven would be two essential items that will meet 90% of your needs when cooking on a stove or in the oven.

If you want to expand the utility a bit more, opt for an enamelled dutch oven to make it better for high-acidity dishes like tomato-based dishes.

A lightweight frying pan made of aluminium is a nice addition to this, and I'd opt for the cheapest non-stick thing you can get. It's not essential but it's a handy thing if you want to do some quick cooking or for things like crepes. Personally I have no love for non-stick cookware but especially for a home cook who is starting out it can be much easier than the alternatives.

I like Pyrex and similar glassware for its utility. The bowls can be used for all sorts of things, including baking and microwaving.

I'd recommend a good paring knife, a good santoku knife or chef's knife, and maybe a bread knife if you're big into bread (especially crusty or artisanal breads). You don't need a big knife set. I'm very partial to Kiwi knives but I hear that they are a bit pricier in the US. Here's an excellent post on knives and the discussion in the comments is worth reading too.

Steak knives are also really good (hear me out) - they're great for slicing things like ripe tomatoes and tougher foods. Not something that I'd expect would be the first thing to spring to mind for a vegan but I've been a vegetarian since forever and I use steak knives all the damn time. If you care about the edges of your cooking knives and you want to do something like slice a burger but you don't want to get another cutting board dirty, you can use steak knives to cut directly on the serving plate without feeling guilty.

A nice pepper grinder is essential imo. I do love absurd amounts of pepper so ymmv but using enough pepper is one of those things that can really elevate a dish. I like one that has a good mechanism to adjust the grind size so you can do things for texture.

This also applies to a nice salt grinder - it might be a bit me-specific but I love popcorn and I hate the ridiculous bagged stuff (that probably isn't vegan anyway). With a big pyrex bowl, something to cover the top like a plate or some cling wrap, and a bit of vegan butter you can make popcorn at home for pennies. With a nice salt grinder you can adjust it to the finest setting to make the equivalent of popcorn salt.

I like having a fish spatula on hand for dealing with delicate stuff that I'm frying and for getting under things that have developed a bit of a crust in the pan.

A heatproof rubber spatula is also invaluable. You can also get spoon-shaped spatulas that are super versatile, like this style:

I love things that make vegetable prep easier too.

The OXO handheld mandoline slicer is really good and it's probably ideal for your purposes. Definitely buy a cut resistant glove if you're going to use a mandoline though.

These julienne graters are really good too - the finer ones make really nice garnishes and it's ideal for salads and similar:

I'd strongly recommend using a cut resistant glove with this as well.

What else?

A ladle is useful, as is a stainless steel cooking spoon, and a nice broad wooden spatula.

I'd recommend investing in some nicer metal measuring cups and measuring spoons once instead of buying cheap shit.

I really like this style of measuring cups because it has internal measurements inside, so if you're making a dish that uses different cup measurements you can usually get away with only needing to dirty one measuring cup:

On a similar note, I don't think you can do better than these magnetic dual-sided measuring spoons:

They have a narrow side for getting into small containers and because they're dual-sided you sorta double up with the amount of measuring spoons you have - if one side is dirty and you need to measure out that quantity of something like salt you can just use the other end.