For the overwhelming majority of military history, the core levy troop structure consisted of spearmen. Spears are more versatile than most other weapons as they allow for closed ranks. Closed rank foot soldiers didn't need to rely upon martial prowess to carry the day, as it was momentum and distance between the combatants which aided morale in the face of an opponent. That spears also resembled farming tools which most medieval levies would have had extensive use with, and that a phalanx of levies was the best available deterrent against a cavalry charge were a given. Confirmation of this is recorded all over the place. A particular example of the presence of spears in the zeitgeist would be the Bayeux tapestry, to commemorate Duke William's victory over the Saxons at Battle, outside Hastings. The presence of other weaponry is noted, but otherwise the artisans of the time period were quite familiar with the gear that men took into battle with them. This gallery of assorted equipment from various periods, including from 1066, is also possibly of interest

Other common features on the battlefield would be archers, and later the more economical crossbow. Hunters trained in fieldcraft and archery were not typically present in such large numbers to benefit from being paid to march into battle, and the conditioning for a young man to reach adulthood with the precision and strength to pull back a longbow took up most of their formative years. Archers typically suffered from chronic back pain later in life due to deformity from great tension on their shoulder blade as they would need to practice with their core body strength and neglect other aspects of their health. The crossbow by comparison provided the mechanism which removed the need to train experienced archers, and thus greatly reduced mobilisation costs.

The ideal of fancy swordplay and other western martial arts was a romanticised image that in all fairness, had been present from the very conception of chivalry, as melee tournaments proved to be the celebrity sport of the era, where named contestants were not unheard of. Soldiers with heavy equipment like swords and maces were incapable of forming tight ranks. The length of the weapon and the arc of the blows inflicted would result in harming friendlies nearby. Duelling fighters spread out over a battlefield makes for a glorified interpretation of warfare, but the reality is that medieval battles could at times take place over days or weeks as the combatants retreated to their encampments after sustained casualties, and cases where glory takes precedence over survival are not at all normal.

I'm going to generalise quite a lot here, so remember that there are always details and exceptions. Also, on the whole the later Middle Ages tended towards greater provision of arms by lords, kings and cities, and greater use of paid troops, including mercenaries, rather than levies in the classic "feudal" style.

The most famous feature of military organisation in the Middle Ages was the requirements for each vassal to provide troops to their lord. These obligations would extend down to the town and village level. The terms of this relationship would specify a certain number of soldiers, and also the type of soldiers. In particular they would specify how many men-at-arms would be supplied. (And also, often the number of horses - a knight might want to bring too many or too few.)

They might specify (at least implicitly) other cavalrymen, archers, mounted archers or other mounted infantry, spearmen, etc.. It seems like these requirements may have become more prescriptive as time went on. The basic unit the "lance fournies" had at its core a knight, armed squire and non-combat squire, but in different times and places would include other troops, according to custom or agreement.

The equipment of a man-at-arms - your classic "knight" - was, first an foremost, a suit of armour, a shield, a lance, one or more other weapons, and a number of horses. This would be the essential equipment for anyone of rank. The details of the panoply would be up to the individual. Equipment might be required for its symbolism just as much as its military effectiveness - eg. it wouldn't do much for your standing if you couldn't afford modern armour.

Some professional soldiers would be outfitted by their lord, eg. bodyguards or castle garrisons. As the Middle Ages progressed, the use of official armouries became more common.

At the lower level, ordinances might require towns or villages to provide a certain selection of troops. These might be provided communally or be the responsibility of certain individuals. In some cases, all individuals with property or income over a certain level would have to own arms & armour, eg. I'm sure I've seen (but can't now find) requirements from England for individuals to own a hauberk alongside certain weapons. As WodensBeard said, the most common weapon was the spear - simple to make and simple to use.

As to the very specific question of how a particular individual would end up as an archer, or a spearman, or whatever, I don't know enough to answer - it would require looking at individual biographies. I can only guess that it happened in all the ways you would expect. People would be chosen for their aptitude, would purchase whatever arms suited or gave social standing, would use whatever they inherited, could afford, or could get their hands on...