@cynesthesia's response is pretty informative on the more economic side of things.

is it the case that people basically know what parts of the crust contain lithium and it’s just a matter of finding the vein

My exact field isn't this and it's been a little while since I took courses on it, but for lithium in particular, there's two ways of getting it: from brines (that is, water which has very high amounts of salt in it, where "salt" isn't just meant as the usual 'sodium chloride' but also as lithium and potassium salts, etc), and from solid rock minerals, spodumene being one of bigger ones economically. Brines are much easier to harvest the lithium from, so that's been the focus of lithium mining for decades. This is the situation in the Lithium Triangle in South America, as well as in parts of China. Processing the lithium from solid ore is more energy-intensive and thus costly, but if the concentration (called the "grade" in geology) is high enough then it can be worth it, and spodumene mining is what Australia does as they have high-grade ore. Why exactly this has happened is interesting and beyond me a little; as cynesthesia mentioned, it's surprising how fast Australian mining has come online in only the last decade.

More generally, ores can form in a few different ways. The most common way (or, at least, the way I was taught; as I said, it's been a while) happens alongside the formation of igneous rocks, which happens at plate boundaries and/or volcanoes. One way it happens is that, in a magma chamber for example, a mineral will crystallize (that is, solidify or freeze) at a higher temperature than the rest of the molten rock in the chamber, and that mineral will then be set in stone, pun intended, and settle at the bottom of the chamber while the molten rock floats above it. So when the whole chamber eventually freezes over, when volcanic activity ends, which could take many millions of years, you end up with a layer of a certain mineral, and that mineral may or may not be useful to you depending on what it contains. Another way that it happens is at the end of a magma's "life", when its reaching low temperatures and close to freezing, what's typically left over is the "rejects" that couldn't fit with the rest of the elements in the magma. It's quite low-viscosity for a magma, AKA it's able to flow quite easily, and so it can spread and inject itself into preexisting rock already in the area, and then it finally freezes and a rock called a pegmatite is formed. Now, many pegmatites will be quite boring and contain regular, boring elements, but a few of them can concentrate rarer elements like lithium. Spodumene and other lithium minerals are typically found in pegmatites.

So if you're looking for lithium ore, what you will probably wanna do is a) find a geological location that has shown to have some kind of magmatic activity in the distant past (and these can be all over the world; the planet has had a LONG history), b) search for pegmatite formation, and c) take samples of it and determine what the pegmatite contains. I assume there's a similar process for searching for brines. The problem isn't finding lithium or even minerals that contain specific elements in general per se, they're kinda everywhere; it's finding ores that have sufficient grade to make them economical to mine, and are close enough to the surface that you don't have to dig a whole bunch to get them.

And, to go off on a complete tangent, rare earth elements for example aren't actually that rare all things considered, there's actually been deposits found all over the world AFAIK, the reason that China has a massive production of them is actually because of :deng-smile:. From this article from sciencehistory:

Between 1978 and 1995 China’s annual production of rare earth elements increased by an average of 40% per year. During the 1990s its exports of rare earth elements also increased rapidly, causing prices for these metals around the world to decline sharply. As prices dropped, competing producers either went out of business or steeply reduced their production, unable to meet the so-called China price.

Beginning in the 1990s companies owned by the Chinese government sought to buy controlling shares in rare earth companies located in other countries. In 1995 two Chinese firms joined with American investors to buy Magnequench. This kind of foreign investment needed to be approved by American government regulators: they allowed the purchase but required the company to keep its operations in the United States for at least five years. Five years and one day later the factory was moved to China.

But in 2010 there was an incident between Japan and China, and China began restricting exports of rare earth elements:

Suddenly many people and companies realized that the rare earths had become pervasive and essential. Governments began to worry about the materials their nations needed for weapons and other important technologies. The high prices of rare earths—and the fear of not having them—made bold solutions seem reasonable. Some people demanded opening up the Amazon rainforest to mining or exploiting resources in Greenland. Other entrepreneurs proposed extracting rare earths from the sea floor or raking them off the Moon. After investing $500 million on new pollution controls for its processing operations, Molycorp reopened the Mountain Pass Mine in California in 2012.

But the boom produced by high prices proved short lived. In 2012 a World Trade Organization grievance brought by the United States, Japan, and the European Union resulted in a loosening of Chinese export quotas, opening the flood gates and lowering prices to near 2009 levels. By 2015 it was once again difficult for anybody but the Chinese producers to make money producing rare earths. The bold plans for obtaining rare earths from around the world were mostly abandoned. Molycorp went bankrupt, and the firm that bought the Mountain Pass Mine now sends its semi-processed ore to China for final processing.

So China played the economic game well, knowing that the neoliberals would neolib, and they've hated China for it ever since. This is a glimpse into what could happen if the West tries the sanctions war on China. But it'll take many years to set up profitable rare earth mines even if the war started tomorrow, and what might happen instead is that the rare earths will go through middlemen to the West, causing them to pay higher prices (and for the middlemen to get rich), kinda like what we're seeing with Russian oil.

Economic geology is quite a political game. It always has been; a century ago it was oil that propelled the desperate Nazis to invade the Soviets, and this time, it might be rare earths and lithium that propel the United States to invade China and South America.

Sorry, this got long. I should write a post on this at some point.

I'm reading more on it and I feel like this is something I'd need a textbook to understand fully, but I think the order of events is that the rift was forming 500 million years ago or so and continued to do so for the next couple hundred millions of years, but it was the failed arm of a triple junction. Then the Variscan orogeny came along and this resulted in further synclinal structures as the crust buckled (so the Azov Uplands are perhaps the eroded remains of some of the mountains formed during that orogeny?). There was further volcanic activity in the area due to the orogeny, and the synclines flooded in the Middle Devonian, forming basins, then the waters eventually retreated, came back, retreated, etc until the glaciers covered the area and eroded it closer to the modern day, which is probably why the country is so flat in places, and here we are today.