Generally, any one-on-one sword-fights in any pre-gunpowder theater of war are an invention of modern cinema. Sure, if you really want to be pedantic, Bronze age cultures did conduct warfare that way (probably) but for the most part, a fighting force is only as strong as it is united and disciplined.

So by and large, pre-modern battles were fought between tight formations with polearms. I'm most familiar with high medieval Italian militias, but I don't think I know of any world culture where mass mobilization existed where this wasn't the case. Battles were principally characterized by infantry slowly grinding each other out. In Europe, opposing cavalry typically engaged each other on one or both flanks in order to try to wrest superiority, and whomever managed to achieve supremacy would try to to encircle the opposition. A horse won't charge head-on into a static wall of spears, but unsupported infantry will break and run if flanked by charging cavalry. There are differing opinions on how the specifics of engagements played out; some argue battles were short and bloody affairs, while others argue two battle-lines would reengage and disengage for multiple short periods over the course of the day's engagement (I'd say it was probably a little bit of both depending on the situation).

It's also important to point out that large-scale battles were the exception rather than the rule. Large, decisive battles between evenly-matched forces were risky business, and most leaders tried to avoid them. During the skirmishes that could occur between raiding or scouting parties, more open and less structured clashes could occur. But records of these smaller clashes very seldom exist. Insofar as the leadership was concerned, medieval warfare was mostly a slow, drawn-out affair that mainly consisted of ravaging, destroying supplies, and laying siege to strategic points. Some of Europe's most famous warmongers fought surprisingly few battles: In the tenth century, the German Kaiser Otto I usurped the crown of Italy without and there is not a single record of a pitched battle; in the eleventh century, William the Conqueror fought a single pitched battle in his whole military career, and that battle made him King of England; the following century, Emperor Barbarossa managed to reassert imperial control in Italy by focusing on sieges and controlling strategic points, and lost everything when the Italian forced him onto a decisive battle.

So how were soldiers trained for this kind of warfare? By and large, soldiering in medieval Europe was the exclusive purview of people who already owned arms and armor and knew how to use them. These two requirements typically meant that military recruits were landowners. Sometimes fighting men were recruited with the promise of monetary reward, other times social and political ties were fostered by granting of some sort of privilege that came with the duty to fight. The wealthiest and most powerful would normally employ retinues of professionals who do not fill the mold of the prosperous property-owning fighting man. These men could be landless knights taken into the household, or merely "Sarjeants," that is to say servants trained in the use of arms. And in all categories, there is a fair degree of overlap. For example, there is an ongoing debate about the composition of King Harold's fighting men at Hastings: newer scholarship argues they were almost all professionals in the service of great magnates, while others argue they were specially recruited warrior-landowners.

In my own field, Italian political organization, there is a fairly unique integration of multiple segments of society in armed conflicts. Although multiple european states also involved diverse urbanites in their armed forces ( places like Portugal and Burgundy stand out, with the latter particularly good at recruiting burghers armed with crossbows) these were compliments to a broader force of recruits consisting of property owners. No other states were as reliant on urban recruitment as those in Italy. This is, of course, a consequence of the urban-centric political organization in Italy: when an urban council like that in Florence or Milan decided to go to war, the citizens who were most represented in the council would be expected to be the first to mobilize and bear arms. For these purposes, "Citizens" generally means "Upper-Middle Class Males," but that would also include most members of the more prosperous guilds, the wealthiest among the members of minor guilds, as well as suburban landowners. On top of this, even though the poorer members of society could not afford to arm themselves autonomously, organizations based around neighborhood parishes or trade guilds would try to pool resources in order to contribute fighting men, and martial training and drilling would generally be in the purview of community leaders, typically members of the guilds or parishes who already had some experience in warfare. In the largest cities like Venice or Milan there were even elaborate systems in place to extract recruits from the poorer members of society: in Venice, lotteries determined who was drafted into the navy, and every draftee was drawn with three neighbors who would pay for his arms and armor. In Milan great lords called Capitani seated in the polity's executive council were expected to provide arms and armor for those who could not afford them in their neighborhood. That being said, even with these systems in place the difficulty of mobilizing substantial numbers of people from the minor guilds and/or lower middle class meant that there was typically a disproportionate amount of upper and middle class citizens present when Italian hosts set out in force. Plus, this system generally made for poor-quality fighting men: Italian armed hosts performed poorly when pitted in the open field against French or German forces whose recruits were drawn from a more traditional and typical warrior-aristocracy.