My husband and I recently built a furnace and have been smelting copper, bronze and even made an iron bloom last week. We enjoy primitive technology and a lot of times when trying to emulate and recreate how our Neolithic, Bronze Age and Iron Age ancestors did things, it becomes sort of apparent how they figured it out, the first example that comes to mind (that has nothing to do with metal) is that cheese was "discovered" when someone stored milk in a bag made of the stomach of an animal, where that milk mixed with the rennet inside of the stomach and became cheese, but unlike a lot of those things, smelting and forging is not an obvious process, it involves a multitude of deliberate actions, some of which are only applicable to the acts of smelting, casting or forging, I cannot believe that some Neolithic person awoke one day, found some malachite or chalcopyrite in the ground and decided to crush it up into a powder, put it in a furnace, heat it to 1085 degrees Celsius and then cast it — so how was it discovered, or rather how do historians think it was discovered?
If you assume that prehistoric people mostly stumbled upon new technology by accident—what I like to call the "dropped in a fire" model of innovation—then yes it's hard to imagine how extractive metallurgy (smelting) came about. However, we have no reason to believe that is actually how it happened. Prehistoric people were as intelligent, analytical, and curious as we are. It is safe to assume, therefore, that technological advances in prehistory were made in the same way the have been in recent history: through a combination of systematic understanding, experimentation, and the occasional chance discovery.
Extractive metallurgy is my favourite example of why the "dropped in a fire" model is wrong, because thanks to some clever recent archaeometallurgical research we actually have a very good idea of how it originated. The short version is that Neolithic people already had sophisticated pyrotechnology because of ceramics, and were familiar with metal ores because they were used to make pigments. It was not a big leap to combine the two.
For the long version, I have copied an old post of mine from /r/AskAnthropology below.
How did ancient Humans discover smelting ores? And how did they get the idea to heat Rocks with Charcoal while blowing air?
Once upon a time people would answer with this question with a shrug and something about leaving shiny rocks in a campfire, but thanks to recent archaeometallurgical research we actually know quite a bit about how it came about. It's my favourite example of how technological innovation—even in so-called "primitive" societies—is embedded in a complex web of cultural and historical contingency, and rarely just a case of people accidentally dropping things in fires (or leaving them out to spoil, etc.)
So although extractive metallurgy (smelting) was invented independently several times, the earliest known and best-studied example is from the Neolithic of Southeastern Europe. It's from there that the idea spread to the rest of Europe and probably also the Near East. But before we get to that let's backtrack a bit. Humans have been been familiar with broad concept that fire can have a desirable, transformative effect on materials for a very, very long time. We were using it to cook food millions of years ago. Later they discovered it worked on inorganic material too: heat-treating flint and chert makes it harder and easier to flake into stone tools. It also changes its colour, and from at least 12,000 years ago people were deliberately taking soft stones, grinding them down, then burning them to produce brightly coloured decorative beads. Finally, around 8,000 years ago, there was a crucial development when people discovered that firing clay produced hard, watertight ceramics. This had actually been discovered several times before, but it really took off at that time because of another recent invention – farming. Pottery made agriculture much easier, allowing people to store food, protect it from pests, and cook it in open fires by boiling or steaming.
At the same time as people were experimenting with fire-treated material, they were becoming familiar with metal ores. Oxides were frequently ground to make decorative dyes, powders and paints; or shaped into beads. By about 11,000 years ago they were also exploiting the small amounts of native gold, silver, copper and (meteoric) iron available on the earth's surface to make beads, jewellery and other personal adornments using "cold" (i.e. low temperature, non-extractive) metallurgy.
But what about extractive metallurgy? Well, when the first farmers of the Near East migrated to Europe, they brought all these technologies with them. And their descendants in Southeastern Europe became very keen on ceramic technology. The produced a wide variety of vessels, but also used it in the construction of their homes, to make floors and furniture, and in domestic rituals centred around miniature ceramic figurines. High quality pottery needs to be fire at controlled, high temperatures, so they developed sophisticated kilns fuelled with charcoal. They were also very keen on decorating their ceramics, and the fashion at the time was for bright, colourful and/or shiny patterns.
So at this point (around 7,000 years ago) all the elements are in place: you have a group of highly skilled ceramicists with sophisticated kilns, who were familiar with a variety of colourful metallic ores and were constantly experimenting with new ways to produce eye-catching pottery. And in the area of modern-day Serbia, this culture happened to coincide with a natural abundance of copper ore deposits. In this context, Radivojević & Rehren (2015) describe how the first step towards extractive metallurgy may have happened:
Although the original reasons for applying heat (as in smelting) to copper minerals are not yet well understood, a clue for such a decision may be found in experimental reconstructions of copper smelting events. The primary author recently took part in a copper smelting experiment and recorded the moment when the flames of this process went green (Fig. 14). This happened just after a handful of malachite was thrown onto a bed of charcoal heated up in a small bowl furnace. Thus, it could be assumed that early attempts to smelt copper were potentially driven by curiosity to explore properties of materials that burned with green flames, and that knowledge of copper smelting evolved in the context of appreciation of the aesthetics of coloured flames.
Our first evidence for it comes in the form of small amounts of slag and tiny copper droplets in ad hoc, but efficient, ceramic-lined pit-kilns. They only used ores with distinctive black and green colours, so perhaps these were the remains of one-off, experimental or preformative episodes like the one Radivojević describes.
Later, as they gained an understanding of the smelting process, they began to use more complex ores and were able to produce a substantial amount of copper implements. But overall they were still rare, and probably inferior to lithic tools in many respects, so presumably they were valued for the exotic and aesthetic properties. Metalworking itself probably remained the closely-guarded and ritualised secret of a small number of specialists – some of the oldest European folk-tales, dating back as far as the Bronze Age, are about smiths using otherworldly knowledge to outsmart devils.
To summarise: as far as we know the earliest use of smelting was in Southeastern Europe around 7,000 years ago. It was invented by skilled craftsman by a process of deliberate experimentation, drawing on a sophisticated knowledge of pyrotechnology, the aesthetic and material properties of metal ores and "cold" metalworking that went back thousands of years. At the same time, it was also the product of a unique and fortuitous combination of existing technical skills, cultural values and natural resource distribution.
Sources:
/u/brigantus, a mod for /r/AskAnthropology made a post about what you are asking here
The 3 processes you list - smelting, casting, and forging - don't need to develop at the same time. This makes accidental or deliberate discovery easier.
/u/brigantus has already discuss the development of smelting, which was also recently discussed in https://www.reddit.com/r/AskHistorians/comments/849n4k/how_did_ancient_civilizations_discover_how_to/ and I won't add further to that.
The earliest smelting appears to have been of copper, and the temperatures required to melt copper (which is required for casting) are not too challenging to reach. Copper smelting is often carried out in the molten state, producing liquid copper as the product.
smelting and forging is not an obvious process
Why is forging not an obvious process? Hitting something with a hammer in order to reshape it seems fairly obvious.
Of the three technologies of smelting, casting, and forging, forging predates the other two by many thousands of years. Before smelting, the material worked had to be found naturally in the metallic state - the most common of such "native" metals are gold, copper, silver, platinum, and iron (either telluric iron, which is rare, or meteoric iron). Prior experience with native copper was probably important for recognising when copper was accidentally produced when making pottery.
The earliest forging was cold-forging - you take the metal, at room temperature, and hit it with your hammer to reshape it. Hot forging only becomes significant with the development of iron smelting, by which time smelting and casting were already very old technologies. Forging is very old (already in use by 8,000BC), but hot-forging is much more recent.
The relative newness of hot-forging doesn't mean that it was difficult to discover. Copper and copper alloys are usually cold-forged. Many copper alloys (like traditional tin-bronzes) are brittle when hot, and must be cold-forged. Copper can be hot-forged, but the forging temperature is close to the melting point, and you run the risk of accidentally melting your work when heating it to forging temperature. Unless you are trying to forge a large thick piece, cold-forging is easy enough, and if you want a large piece, those can usually be cast. Until you are working large pieces of iron, you don't need to develop hot-forging.
There are multiple possible paths to developing hot-forging: annealing of copper alloys and welding are two of the more likely. When copper alloys (and iron) are cold-forged, they work-harden. If you continue forging after they have work hardened too much, they will crack rather than deform. To be able to continue forging, you anneal the workpiece. The piece is heated and usually quenched in water to cool it. If a smith cold-forging iron does this, and experimentally hits the workpiece while it is hot, they will find that it is much easier to forge.
Welding by heating the two pieces one wishes to join and then hammering them together would have led to the discovery of hot-forging, if it had not already been discovered - the workpiece will deform during welding.
Annealing and quenching of copper and copper alloys to remove work-hardening might have been inspired by the heat-treatment of rocks to improve workability for making stone tools. Otherwise, it would be simple enough to observe that copper, as cast, is initially easy to forge, and make the connection from prior heating to ease of forging.