And how did such knowledge get so widespread? Were there just surface deposits of metal that someone decided to play with one day? It just seems like such a complicated process to have just... happened.
And what about certain weapon types, like the Axe or the Bow? Like the native americans and europeans/asians seemed to use several of the same types of weapons. Axes, Swords, Bows, Spears. The spear I understand, but how did the rest get so widespread among cultures that hadn't found one another?
The general belief among historians of science and technology is that first metals that were used were gold and copper. Both of these occur as nuggets of almost pure metal in nature, and they are easy to distinguish, easy to work with and very resistant (or immune) to corrosion.
The next steps are more uncertain, but the most plausible route towards smelting seems to have involved cooking fires accidentally set up with ore-containing stones. Pyritic copper ores (i.e. primarily admixtures of copper and sulfur) in particular can be fairly easy to smelt, even in a cooking fire. It is generally thought that people began to notice copper among the ashes of a fire, and began to experiment. Relatively simple methods could and did produce copper from rich ores even up to Roman times and afterwards.
Human prehistory is long and there must have been multiple opportunities for many technologies to develop. It should also be remembered that even though our popular understanding of technological development gives undue weight to singular discoverers - in my personal opinion, because it makes for a better story - the truth is that simultaneous or near-simultaneous discovery has always been the norm and still is. Most inventions seem to have been invented by multiple persons who don't have any contact with each other almost as soon as the prerequisites and a need for the technology exist.
Therefore, axes were likely invented many, many times by people who noticed that attaching a lever to a hand-axe (blade) would make it far more effective, and bows by people who (perhaps) took notice of how a sapling could store energy when bent and throw a rock or something quite a distance.
As I said, this is still the norm these days: invention occurs when the conditions are right, and usually multiple people make the same discovery within a short timeframe. My favorite story is the patenting of telephone - patent applications, most likely resulting from independent work, were separated by IIRC three hours - but we can see the process in the development of social networks too.
For more information, I may suggest Basalla's The Evolution of Technology (1988).