The resource options are a good point, along with the conversation on who fought with what.
I did a ton of research on this when I was making my Ironworks mod for Dwarf Fortress. So I know a decent amount about all of this.
Generally speaking, cultures used whatever was most effective for fighting the other cultures they came up against that was still cost effective.
The Japanese, for instance, had fairly light katanas that were designed for fighting lightly armored opponents. When they had to fight the Chinese/Mongols, they ran up against more heavily armored opponents and had to create a heavier sword with an iron core to help them penetrate the armor. In general, though, they were an iron-poor island that mostly just fought itself, and much of their fighting style was centered around that. Katanas have a legendary reputation because they were, in fact, made to very exacting standards. They had to be, because a poorly made katana would break easily and a broken sword is pretty useless. So they're really sharp and well made, for the most part, but not actually 'better' than a well made western sword. And certainly not 'better' at fighting an armored opponent, since that wasn't really what they were designed to do. Katanas are slashing weapons, and trying to cut a guy in plate mail is pretty useless, especially if he's got mail underneath.
Swords in mainland asia and the west were made much easier since decent quality iron was more abundant. Most 'iron' weapons are actually a very low grade steel, since pure iron is pretty worthless for weapons and some carbon almost always ends up in weapons during the forging process anyway.
India did, in fact, have the first steel alloys, due to some fortuitous iron deposits that contained impurities of Vanadium and Magnesium. This is why Damascus or Wootz steel was so valued, it was something that no other region at the time could replicate. During the later roman empire and the middle ages actual pattern-welded steel forging techniques became pretty common, but even those weren't as good as the steel alloys the Indians had. Before that, Bronze was going to be better than the vast majority of iron weapons but was both expensive to make and difficult to work with compared to iron.
Eventually those deposits dried up and India was back around the same tech level as everyone else.
From a game perspective, noting how easy it is to get various resources should help determine what tech level people are at. If you have ample deposits of Copper and Tin, bronze should be the favored metal for a very long time. Otherwise, Iron is more abundant in the world and will generally supersede bronze because it's cheaper and 1000 guys with iron weapons will probably beat 100 guys in bronze no matter what quality you're using.
As far as tech goes, people use what they need to and rarely beyond that. If all of your neighbors are using stone spears, you probably don't need much better than simple metal weapons and leather armor to beat them. If they've got chain mail and decent quality metal weapons, you're going to need some decent equipment of your own to beat them. If building an army strong enough to beat someone takes more men or equipment than you can afford, you probably won't do a whole lot of invading.