It really depends on the circumstances and the exact medieval period. I don't want to delve too much in the origins of the word 'sergeant/serjeant', since it used to mean a lot of different things.

For instance, knights usually were maintaining lances - not in the sense of a weapon, but of a prototypical squad, a retinue. Besides the knight, who was both the commander and the "elite warrior", such a lance included one or two squires or men at arms, a small group of archers or other low-born peasant levies. The Knight from Canterbury Tales for instance has a squire, who is his son, and an Yeoman. One of knight's squires could be a non-fighting one, commanding the noncombatants (servants, grooms, blacksmith, perhaps even a literate clerk, medic or priest). The lance usually wore knight's colors, and if possible, used best weapons available. However, peasants were rarely drafted, since less men meant less farmhands and less food.

In such an arrangement, knights could serve as sergeants of sorts, but the units were smaller. Drill - beyond maybe weapon instruction - was nonexistent. While Roman Legions were highly trained to cooperate, each knight, as seen above, was basically an unit by himself. Knights either learned fighting from their fathers or during their time as squires, although later professional instructors and manuals appeared.

Beyond maybe basic training, peasants didn't receive much instruction. Peasants were usually equipped with spears, pikes or at best bows, because unlike swords they required little training. (In England everyone had to practice archery, and that's why English archers were so famous.)

Were the combat instructors harsh, violent and abusive? Quite likely, given that most instructors were harsh, violent and abusive in that period, whether of fighting, craftsmanship or reading or writing. It was only in the twentieth century when beating stopped being considered an educational aid.

The Crusades, crossbows and later gunpowder changed that and large armies became more popular from 14th century onwards. The condottierri in early Renaissance Italy certainly had sergeants in their large mercenary companies.

A reply to /u/kwertyuiop

In early modern era, armies of the time started to formalize training of recruits. I wrote about this here.

The Spanish expeditionary armies in Italy and Flanders developed a system where fresh recruits from Spain are sent to Italy for some period of garrison duty and training, before they sent to Flanders. There was also an officers training program. They were the best system in Europe in that era. It is no surprise that the Spanish gave us the root words for infantry and sergeant and many other military terms.