How do people find mining sites?

Exploration. There are lots of keys about what might be in the rock formations, such as the type and age of the rock. The vegetation that grows in an area can also be key as they take up loose elemental metals in the soils. What a company will do is take several rock cores over a grid area and analyze them for the contents to determine where it's financially viable to dig further.

they work their tails off looking. what specifically is done depends on what is being looked for and how much money the person (or corporation) is willing to gamble. The main idea is that they look for generalized geological features that should be associated with whatever they are looking for. that requires looking at all geology to define how things actually are. the simplest form of prospecting is to just go out and sample things that look like they fit what you want to find (say, looking for quartz-carbonate veins if looking for certain base metals or gold). Frequently, more complicated geological, geochemical, and geophysical methods are used to narrow down the target zones to manageable (explorable) areas for detailed exploration (geochemical surveys, gravity or electromagnetic surveys, etc.). There are a very large number of technologies that may be used, and many different strategies that may be used for the search effort. Too many to discuss here.