Your character's experience would depend on his background and socio-economic status prior to joining an army. Notice I said "an" army rather than "the" army, as a full-on army would be something only in existence on very few occasions. Is your character a man-at-arms? An archer? If he's a raw recruit, it's unlikely that he's becoming a mercenary; the stereotypical medieval mercenary was more likely a former soldier in a more regular force who could not or did not go back to civilian life.

Either way, the common soldier would generally not be a peasant. By the time of the Hundred Years War, the "typical" medieval soldier and the backbone of an army would be the middling ranks of society; neither knight nor peasant. Many knights and aristocrats never would see combat (or very little of it), but those who did would be largely indistinguishable on the battlefield itself from the men-at-arms. Men-at-arms were non-aristocratic soldiers equipped with heavy armor just like knights would be, and could fight equally well on foot or on horseback. In England, the next rank down from the men-at-arms would be the archers. While the poorest men of the army would be archers, it wasn't necessarily a low-status position. Many families who could afford to equip their firstborn as a man-at-arms would send their other sons to battle as mounted archers.

If your character came from a family like that, he has some training and possibly some very high-quality equipment. He is an ideal recruit for the king's service when England is at war. There's no such thing as mass conscription in the HYW, so he would either be contacted by someone raising a company or he could simply travel to the assembly point and join the army himself. The king's clerks would probably prefer you to sign a contract alongside a company, so that they have less paperwork, but it's not unheard of for individuals or very small groups to be contracted. At this time, practically all soldiers in an army were recruited by contract. These contracts had very specific terms that dictated length of service, rate of pay, compensation for lost horses, amnesty for previous crimes, division of spoils, etc. A soldier would either sign his name or, more likely, make his mark. If part of a company, his captain would sign for him.

There was no such thing as an official medieval officer corps, but armies were not lacking in tactical-level leadership. A company's captain is the officer who would be commanding small units in battle. The captain is undoubtedly an experienced soldier and someone of means, with enough wealth (possibly looted from previous campaigns) to entice. From all available evidence, it seems that soldiers from prior campaigns eagerly flocked to the staging point whenever a new army was being put together. A new soldier would have plenty of experienced men to guide/bully him. Companies and retinues would be subordinated to the command of different officers as the situation merited. A small company used as a raiding force one week might be assigned to a larger unit for major field engagements the next day.

Unfortunately, our sources rarely delve into the tactical organization of every level of the army. The senior commanders would be nobility, generally with connections to the royal family. A medieval army was usually divided into three divisions, with skirmishers deployed as the terrain allowed. Special groups might be organized for specific tasks, like outflanking or handling prisoners, but these groups would be improvised and dissolved after the fight was over. As you might guess, this was a very different method of organizing for war than in modern times. Political crisis could severely diminish the army's ability to fight, since the commanders would also be some of the most important people in the kingdom. Ineffective political leadership could utterly squander even the most experienced and skilled armies.

You're probably going to have to be a little more specific than just stating "medieval times", since that covers roughly 1000 years, and applies to a bunch of states. Are you envisioning a Western Europe type during the First Crusade, or a Eastern Roman Empire type during the Komnenos Restoration, or the Arabian Mamluks, or the Golden Horde, or Italy in the Renaissance?

Provide a specific time and a country, for start.

Did they have anything comparable to sergeants?

The modern rank of Sergeant gets its name from the French Haigh Medieval Serjeant. The exact privileges and role of the Serjeant changed through time, but it was generally a middle, land-holding rank somewhere below an enfeoffed "knight" and above a tenure holding farmer.