Well, in the case of South America, there were certainly some large-scale battles between the various complex organized societies, but it's hard to get the meat of their longer overall history, because their civilizations were completely laid waste. They did in fact keep records, oral and written, but the written records they had are rather cryptic (Quipus) or deal more with religion (think Mayan Calendar). We do know some things about the ones who faced the Spaniards (Aztecs, Incas, etc), but that's pretty incomplete. We know very little about their predecessors or fallen enemies (Olmecs, etc.)
Africa had very large settled empires at times (you know Magic, so if I say "Mirage/Visions weren't actually that bad a fictionalization of African civilization circa 1400), but has suffered disproportionately in the past five hundred years so much of the records are lost. Africa's history actually came up recently in a discussion elsewhere about "Why is modern Africa such a shithole exactly?" (answer: Several factors, both domestic and external but the Europeans and slave trade were the largest. Just a shitstorm all 'round), so instead of summarizing it myself, I can re-post someone else's much better summary for your edumafication! Apologies if there are loose references anywhere - I excised all the irrelevant discussion bits and just copied the posts with the information.
Africa in the 14th and 15th century was politically and technologically not much behind Europe, if at all, especially on the Western Coast. The political and technological devolution of Africa came about for a lot of reasons, some domestic, but by far the biggest catalyst and the reason for how things got so shitty was the trade for slaves (as exports) and guns (as imports), in conjunction with the tsetse fly. The latter meant that Europeans couldn't spend much time on land and work out a deal with the Songhai or some other major kingdom, propping up puppet rulers to make trade easier. Instead they had to rely on coastal trading fortresses (which still had a very high mortality rate even for the white traders), which meant that the most profitable dynamic was to form a roving gang of men with guns who went around kidnapping people and selling them into slavery for more and better guns. These conditions aren't conducive to any kind of stable political order, and a result most kingdoms and political states founded after the 16th century and before colonization had a very short shelf-life.
Then of course medical advancements allow Africa to be colonized in the late 19th century, and by that time most of Africa is a stateless, tribal and highly paranoid, nomadic place. European anthropologists come along of course and draw lines on maps and decide this is where such and such "tribe" has always lived and always in these conditions, because it's not like there was civilization here once or anything crazy like that; these are black people. Colonial treatment of native populations ranged from the merely brutal to the Belgian*, and for all the talk of "civilizing" Africa there was precious little effort to industrialize the place at all; then as now it's just a place for exporting raw materials and importing unquestioned aid. The only difference is that the despots are closer now and we feel less guilty about it because we're not directly exploiting the place.
So Africa sucks, sure, but the slave trade, both directly and because it started the meme that blacks were a mentally inferior race to be subjugated, has a lot to do with that, so it's kind of ridiculous to say that captured slaves were "lucky" because their ancestors would eventually be better off than people living at bare subsistence in a despotic wasteland.
*King Leopold was basically the prototype for Himmler. In fairness this was too much even for most Europeans, and there was wide condemnation of the activities in the Congo, Arthur Conan Doyle being one leading critic. It was the basis for the novella "Heart of Darkness", and one of the reasons why the Congo is so shitty today, even by the standards of sub-Saharan Africa.
I am confused. Wasn't the reason the Europeans started capturing and enslaving blacks that we perceived them to be inferior due to their culture and lifestyle being so primitive? If the Europeans arrived there and saw that they were advanced and enlightened, why would the Europeans have brought out the nets and collars?
Well, it's a bit more complex than that.
Slavery was common in Europe well into the medieval period. It was repeatedly banned by the Church, though they could not enforce the ban and it was unsuccessful. There was some success in banning the sale of Christians to non-Christian nations (mostly to the Muslim world, I think). Meanwhile, the established African kingdoms of the time had forms of slavery that varied in severity, but mostly resembled indentured servitude. Finally, Arab slave-traders had been operating in Africa for hundreds of years.
All of this is to say that all parties had some experience with slavery and it's infrastructure. EDIT: I see Xyre got that bit already.
The "slavery" you're thinking of (chattel slavery of black Africans by white Europeans) really took off in in the fifteenth century. The Portuguese discovered that the mid-Atlantic islands they now owned were great for sugar plantations, but couldn't get any settlers out there. So they purchased African slaves from the gold coast.
After that, the Pope made a landmark declaration that allowed the Portuguese to reduce Saracens, Africans and the like to chattel slavery. Armed with a moral blank cheque, the Portuguese and Spanish proceeded to build large scale slave-based economies in the Atlantic and in the new world, where (again) attracting settlers was very difficult. At the same time that the church gave the a-ok rubber stamp to enslavement of darkies, slavery of white Europeans by other white Europeans was increasingly frowned on. In order to justify the widening dichotomy, racial and religious justifications began to creep in, later followed by pseudo-scientific ones.
Admittedly, this is not one of my areas of expertise, but the important points are that Europe was at least familiar with coastal Africa prior to the age of exploration and that the African kingdoms (while hurting) really didn't collapse completely until the sixteenth to eighteenth centuries.
****
Only somewhat related but interesting all the same: Did you know most of the Italian Renaissance was paid for by a huge amount of gold that was swindled from the Kingdom of Mali? As usual, the guys on the short end of the stick had no idea the Europeans valued the shiny stuff quite so much, though in that case the king (Mansa Musa) was pretty profligate.
The first slaves traded in Africa were like the first slaves traded anywhere; criminals, political dissidents, undesirables, victims of war. And they were traded by the African powerful elite just like they were traded at the time by the European and American and Asian powerful elite.
The reason that African slaves became en vogue had nothing to do with an initial perception of inferiority; the idea that they're an inferior "race" is a somewhat new meme, historically limited to the past handful of centuries. It was simply the fact that sub-Saharan Africans, due to exposure to the tsetse fly, were much more resistant to disease than other slaves, and could work harder and longer in the tropics due to this (the native populations of most of the Caribbean islands died off almost entirely within a few decades of the Spanish arrival, being far more vulnerable to the diseases the Europeans brought).
It was increasing demand for African slaves, coupled with the introduction of quickly advancing firearms technology and domestic fragmentation along ethnic and political and religious lines that caused Africa to slide backwards, really reaching a fevered pitch of disintegration after the collapse of the Songhai Empire in the late 16th century. The Songhai were pinched between domestic balkanization (at their height they ruled an area around twice the size of France), invasion from Morocco and the fact that as demand outpaced what had been the "legitimate" supply of slaves for Caribbean plantations, European traders were basically arming roving gangs of thugs with technology that the existing powers couldn't cope with on top of everything else. It's like some aliens set up shop in their UFOs over the sky and started offering laser guns, jet packs and force fields to anyone who brought them captives.
European powers probably would have tried to conquer these places directly, or prop up the existing structures that would become indebted to them, but Europeans that ventured into mainland Africa had a nasty habit of dying very quickly and painfully from exposure to diseases they couldn't cope with. The result is that the entire thing fell apart. By the time of African colonization in the 19th century, most of Africa really was a stateless society, and it was simply to European advantage to pretend that this was an inherent condition of being African rather than the result of specific forces in history for which they had some significant blame. Most of them also probably didn't know better.
Basically, a line graph of European and African economic development over time shows Europe ahead up until the Dark Ages, dramatic collapse in Europe, Africa continues upwards, they're about equal for a while, Europe begins pulling ahead a bit after the year 1000, but up until about 1500 they're about the same level. Then Europe surges up dramatically and Africa takes a long dive that continues pretty much unabated up til the modern day. In real terms the average sub-Saharan Africa, outside of the Southern Cone, is about as poor or poorer than they were five hundred years ago, and HDI is almost certainly lower than it was under the Songhai.
As for the other two continents, some of the North American hunter-gatherer groups were trending towards settlement and agriculture - and at times they started to have very structured agrarian societies, but most "battles" were still little more than large skirmishes or raids. I don't know too much about Australian aborigines, but they were never very thickly populated and had very little agrarianism (if any), remaining almost wholly hunter-gatherer. Cannibalism was by all accounts popular, but beyond fights with a handful of warriors (again, skirmishes or raids) it's extremely unlikely that anything would have taken place that would be of any scale you could call a battle.
One exception to the last one though would be the Musket Wars of the Maori, but those took place just after European contact in the early 19th century.
