Despite common beliefs in pop-culture, this kind of broad peasant levy wasn't that common. It did, occasionally, happen, but by about the 12th and 13th centuries, warfare in western Europe tended to be waged both by vocational warfighters - knights and the military aristocracy generally - and by hired professionals in the form of mercenaries, often supplemented by hired locals in the form of militias or specialist tradesmen or other workers. Local forces could often embody to oppose invading armies, often by countering raids and other forms of harassment.

This was possible mostly because by roughly the early 13th century, it was expected that many villagers and farmers would have had skill at arms, and would have been pretty experienced as wrestlers, would have had the right to bear arms, and would have been part of the local village or town's militia structure. These were ubiquitous, and while they tended to explicitly empower an in-group (usually local property owners or leasees) while preventing out-groups (transient workers, day-laborers, paupers, foreigners) from involvement. The actual structures and their day-to-day operations would be wildly variable, because towns and cities all had local power structures and localized interactions with higher political or religious authorities. A city with a university has to deal with a rambunctious student population (an out-group with money and political power who are nevertheless "foreign" to the city in which they study), and a town on the "marches" of a polity might have to deal with foreign raiders, and towns in the Holy Roman Empire might be subject to property destruction and violence as a result of feuds.

Things can still manifest in relatively straightforward hierarchies. Independent cities - "free cities" - had sometimes essentially purchased their right to self-governance from their lords and could govern their own affairs. That meant, generally, that it was a thorny question as to whether or not the lord that has the right to tax the town or city also has the right to compel their citizens or townsmen to labor or military service. This was another local issue, and could manifest in extremely complex ways in individual cities. Sometimes, whole conflicts were waged over questions like this; in the early 16th century, Duke Ulrich of Württemberg waged a war against the city of Reutlingen after citizens had, allegedly, killed Ulrich's appointed forester, Ulrich made it clear that he expected to oversee their trial himself. The citizens disagreed that he had the right to oversee the trial, and appealed to the Swabian League to defend them.

Cities tended to be very protective of what they perceived as their rights, and maintaining a credible and effective militia was part of that. They also had deep pockets and could raise money to purchase professionals as well, and so even in times of great danger, towns were unlikely to round up any nearby citizens and force them into a shambolic army. They embodied the best, most practiced, most proud citizens and made it a point of civic pride that they could field an army. The Free City of Nuremberg not only brought a force of citizens to fight in the 1499 Swiss War, but also fought against the regional nobility in 1502, which culminated in at least one quite intense pitched battle. In 1504, the Landshut War saw free city militias, mercenary forces, volunteer aristocrats (Like Götz von Berlichingen, who lost his hand in this conflict), and foreign armies fight a succession war. Militias fought against massive armies in the Italian Wars, and sometimes even staged stupendous upsets, such as the 1500 siege of Pisa. More common was the simmering harassment of locals opposing the "foraging" of invading armies. Charles the Bold's forces during the siege of Neuss were harassed by local peasants and driven away from important ground. Charles also had to contend with mutinies among his mercenaries, which was a touches on another question you had, which we'll get back to below.

Obviously I am focusing here on the very late middle ages, but similar social systems and expectations had existed for some time, all around western and central Europe. Widespread drafts of unskilled peasants were uncommon, precisely because unskilled peasants would have made for poorly motivated and poorly performing troops; a liability and an expense in warfare. But towns and cities often promoted and jealously guarded their right to arms, and the ability to pull volunteers from a population of skilled and motivated warriors for mercenary recruitment and for militias is a big reason why warfare took the forms that it did by the 14th century or so. There would have been a huge population of men who grew up wrestling and fencing, who were at least somewhat used to using polearms, crossbows, or firearms (whatever local flavor of any of those were most popular) who could be enticed to sign up for a stint in a mercenary company, or might be induced to serve in a local militia or for a short campaign in foreign lands.

But this all came with the requirement of being treated in a manner that respected their station, and their rights a privileges. You ask if there was any chance to dodge the draft, and of course there was, because the process wasn't a general draft, it was a request for volunteers. You could just not volunteer. But, once volunteered, you could also petition for your rights to be respected, to be paid, to be fed, and to be armed as your contract specified. While mercenaries are often regarded as poor warfighters and have a reputation for indiscipline, mutiny, and treachery, it is far more likely that these acts of collective opposition to their leadership was a direct result of lack of pay or that their leaders violated their contract. A great many atrocities took place as mercenaries taking their pay from the local cities or towns nearby because their leaders were unable to scrape up the specie to pay them. In this sense, rather than the inevitable expression of an inefficient military force, it's a particularly hideous form of a pay strike. And to be clear, here, sacking cities to take your pay was seldom the first action of indiscipline, but the result of repeated demonstrations of dissatisfaction with the situation, until the men finally had enough.

This hits on a couple of your questions; yes, you could oppose your leaders effectively through collective indiscipline in a wide variety of ways. You would likely be pretty plugged into regional politics and know some of the political situation that demanded civic participation. You would have had an upbringing that encouraged your fluency with arms, from crossbows and firearms to polearms and swords. You would often have been responsible to furnish your own arms and armor as part of your service in the local militia and firewatch, and could be fined for not having the right kit. You would have been trained and drilled "on the job" as you marched, and any special tactics could be drilled and practiced in the same way.

By the late middle ages, men drawn into mercenary companies were often pulled from cities precisely because they had this tradition of bearing arms, and because the large populations of cities enabled armies to recruit large numbers of men in a short amount of time. A similar process would take place among the aristocracy, with influential knights or famous fighting men capable of raising dozens of armed, mounted men and retinues to join specific campaigns. Released from house arrest in 1542, Götz von Berlichingen was able to recruit hundreds of men to ride with him on a pair of campaigns under Charles V. These men would all have been volunteer aristocrats and men from their households, who saw violent service as a right and a duty of their station.

I've written about this subject pretty frequently.

How were condottieri hired for wars?

Should I equip my army with bows or guns in the 1600s?

I'm a recruit in an army of the 15th or 16th century. What weapons do I use?

the structure and leadership of civic militias

And for a little on some of the tactical expressions of this system, could men be trained to duck musket volleys?

One period where peasant levies were actually used was in Early Medieval Medieval England, when the Fyrd militia was instituted by Alfred of Wessex from the 870s onwards in response to the Danish threat. The Fyrd was essentially an extension of the existing English system of the gesith, but extended far further down the social hierarchy than had previously been the case. Until the 870s, warfare in England had largely been the province of the thegns and, to some extent, Freemen. Thegns were a essentially a military class; nobles who fought in the warband of the king and were rewarded with prestige, land and gifts - King's thegns - or, as kingdoms grew in size, lesser nobles who fought in the warbands of the King's thegns or of Ealdormen and received similar rewards in kind. These elite armies were highly effective but simply too small to deal with the vast manpower demands imposed by Danish manouvre warfare and strategic raiding in the late 860s and early 870s.

The Fyrd addressed these manpower concerns by extending military service obligations beyond thegns and Freemen to generally 'free' and 'semi-free' tenant farmers alongside their regular service rent obligations. What this meant in practice was that, where a 'thegn ordinary' might previously have gone to fight in the gesith of his King's Thegn, Ealdorman or even the King himself, he would now be expected to bring a gesith of his own. These warbands were organised at the Hundred, or even town and village level, before coalescing at the Shire level to stand as semi-permanent garrisons at fortified burh sites from which they could deploy rapidly against threats.

We know from sources like The Battle of Maldon and the Anglo-Saxon Chronicle's account of the Battle of Brunanburh that within a wider Fyrd deployment, local warbands fought together and indeed could be individually distinguished: after the Battle of Brunanburh, King Æthelstan commends specifically the bravery of the Malmesbury contingent within the Wiltshire Fyrd forces in the West Saxon element of the English army. Those sources also tell us that the Fyrd fought as a disciplined shield wall, and that its thegns and Hundredmen leaders fought in the shieldwall alongside their men, leading and inspiring from the front. Being a member of your thegn's gesith could be a prestigious role, and a thegn would need to know he could depend on the men going into battle around him, so it's likely that your 'recruitment' would be based to at least some extent on interpersonal relationships; you would be known to your thegn or hundredman as a physically capable and reliable person.

How long would I have to prepare and what do I need to be bring? (E.g. "my fathers sword" or do I get all my tools of war from the military as soon as I show up)

As a fyrdman, it's most likely that you'd be expected to provide your own spear and shield. You'd likely carry a seax with you on a daily basis just as a matter of course. We do have some instances from surviving wills of individuals bequeathing weapons and armour to equip specifically ships crews of 'lithmen naval levies, so it's entirely possible that wealthy thegns or other individuals may have provided some level of equipment to their warbands to achieve some degree of unformity or an edge in battle. We have no direct evidence that this did happen, but it's entirely within the bounds of probability that it could have. We do have accounts from the Anglo-Saxon Chronicle, however, of fyrd detachments looting weapons and wargear from defeated Danish warbands and raiding parties.

• Is it expected that I know "how to fight" or will there be training for regular peasants like me?

As a semi-standing force, the fyrd would have had detachments on duty at burh garrison sites throughout the campaign season. We know from 'The Battle of Maldon that leaders gave instructions and advice to their men before battle, and a shieldwall inherently requires an element of discipline, training and coordination to work. While, again, we have no direct evidence, it's probable that fyrd detachments on duty at burh sites would have spent at least some time drilling and training for battle.

• Any chances to dodge the draft?

Well your thegn or Hundredman might be quite surprised and offended at you refusing the honour to join in his gesith, but then your refusal might might be a good sign that you were weren't the kind of person who could be depended on in battle. Thankfully there are plenty of other civil defence service obligations that you could be liable for instead of direct military service. Some of these are detailed in a 10th Century document known as the Rectitudines singularum personarum. Instead of fighting, you could be liable to man a signal beacon or watch-tower inland or stand a coast-guard, to maintain roads or bridges, to build and repair burh fortresses and local strongholds, or simply to produce and transport food on land designated to provide for the burghal garrison.

• and, building on that, how do I get motivated to join the war - loot? salary?

"The Danes are coming to burn your house down, steal your goods and carry off your family" has historically been a fairly effective motivator to violence. Fyrd service was a seemingly prestigious role; Strickland's Anglo-Norman Warfare contains an extensive look at the extent to which lithmen levies became high status individuals within the coastal communities they defended. Military service could also offer substantial material reward. The Anglo-Saxon Chronicle reports a number of instances in which victorious fyrd detachments seized loot, weapons and wargear from defeated Danes, including the Luton fyrd capturing 'much treasure' from the Danes at Bedford in the 890s, and a Mercian army capturing a significant herd of cattle from Danish raiders persued through Cheshire. In reward of the the bravery of the Malmesbury men mentioned earlier, King Æthelstan conferred Freeman status on the townsmen in perpetuity, a significant financial and status boost.

You might also be interested in a question I asked about raising armies in Hundred Year's War era England, with answers by u/sunagainstgold & u/hergrim