From my research, there is lots of information on how to extract ore, but I can't find any information on how these ore deposits are found in the first place, apart from 'looking for surface formations'. So, what kind of formations would exist for different types of ore? Are there any other ways of finding ore?
That is the multimillion dollar question! "looking for surface formations" is indeed one way, and it was the main method of exploration in the past. This does not necessarily mean that you directly see the ore deposit in front of your eyes. Many ore-forming systems are accompanied by different kinds of alteration (for example potassic or argillic alteration) that are not too hard to see in the field. Once you see them, there is a possibility that an ore deposit is nearby.
Other surface methods include, among others, looking at rivers sediments. Since rivers transport soils and rocks from upstream, sampling the sediments may reveal an anomaly in a certain commodity of interest that may lie somewhere upstream. It is also possible to use satellite data - by looking at certain wavelengths it is possible to estimate the mineralogical composition of the surface. This may give hints on the possibility of mineralisation. I've even seen cases where people used different kinds of vegetation that grows on different lithologies!
Surface methods, however, require that the mineralisation will be close to the surface, and the truth is that most of the ore deposits that satisfy that requirement have already been found. This is where geophysical methods come into place. Gravimetric, magnetic and seismic studies probe the earth below the surface and may provide exploration targets, which are then drilled to see what's actually in there.
Mineral exploration is a hard business with much more failures than successes. That said, cases of serendipity are not unknown in exploration. Also, changes in global economics, production costs and increasing/decreasing metal prices may change what is actually an economically feasible ore deposit. The metals are always there - but is it actually worth taking them out?
In addition to the above, what happens is that people look for commonalities between known deposits. So, for example, if you have a lot of gold veins in one area, and lots in another area, then you look at what is needed for these veins to form.
Once you have a reasonable idea of possible ways to form your ore, then you have something that you can look for. Maybe the gold is associated with granite of a particular age range or with particular trace element composition? So if you go somewhere else, and find granite of the right age and trace element composition, then you have a reason to suspect that there might be gold veins in the area.
This methodology is why you will often hear about "types" of deposit. Often these are named after a particularly prominent type-example, such as Missisipi Valley Type deposits for Pb and Zn. Others are named after important features that are present within the deposit, such as Iron Oxide Copper Gold deposits.
You can therefore narrow down what and where you might find things, and use all the stuff that @Michael mentioned in his answer as tools.
Hope that this also helps some.
I've been looking for these things over a few decades and along with other geologists doing this we have seen how we find things change somewhat. Many methods used a century ago are still in use, but with the addition of refinements. So looking at how large and small deposits were found will change over time. (Simple example of that would be that Romans and their contemporaries didn't use geophysics, but they did find major copper deposits in Spain.) Deposits are getting harder to find. It has been estimated that "most major outcrops in the world have been visited by a competent geologist".*
But I must admit, I spend enough time in remote locations to question the statement.
Give prospectors credit for going out and banging on rocks. I know this sounds simplistic but prospectors did develop good intuition about where to look even though they did not apply the genetic models we like to work with today. Ultimately, getting out in the field and seeing what you could find was key. There were also a lot of part time prospectors who were primarily out to hunt or trap.
It takes several rounds (or companies) to usually find a deposit. They are hard to find, and typically you run out of investment and luck before that happens. There are various estimates on how many tries it takes on average, but looking at prospective areas where there have been several in there before you is not a bad idea.
A third thing that can help is revisiting an area that was previously abandoned for a reason other than geology. This could be a small mine that lost its workers to natural disasters, wars or other economic conditions. The previous work may have stopped because the exploration was in the "too hard" category. This is good areas to revisit - particularly in improvements in technology have opened new geochemical or geophysical methods that can tip the balance from too hard.
I throw these three things into "how to find deposits" because even though they are not specific technical methods, they are very much the kind of things exploration geologists think about when selecting areas to use their resources.
* In: Achterberg, F. - Geomathematics: Theoretical Foundations, Applications and Future Developments