It is rather difficult to answer this question using modern state-institution model. There was no uniform system of recruitment, or single army structure, nor was there a single rule that applied to all the Italian states; instead there multiple methods of raising armies that depended on the situation at hand. Every conflict presented unique challenges, which the Italian stated handled differently at different times.

Generally, when a city 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 bear arms. For these purposes, "Citizens" generally means "Upper-Middle Class Males;" so that is to say the majority of members of the more prosperous guilds as well as landed magnates involved in city politics. However, this "standard" socioeconomic recruitment composition could vary: in Milan the upper classes were given a lot of responsibility, while in Florence warfare was very much a middle-class concern.

In addition, even when looking at the Italian urbanocentric political system it is important not to exclude or underestimate the power of the great landed magnates who commanded large retinues willing to take up arms in times of crisis outside of traditional urban organizational structures. Sometimes, these influential individuals could even transcend and dominate urban institutions, building formidable power bases; an example of this would be the Ezzelini dynasty in northeast Italy, who were elected heads of the council ("Podestà") in the cities of Vicenza and Treviso while also holding large fortified estates in the countryside. But powerful magnates could just as easily develop adversarial relationships with the institutions of the Italian city states; an example of this would be in Verona, where the Montecchi, powerful landed magnates based around their namesake Montecchio Maggiore, sparked intense arguments in the city council that devolved into armed confrontation with the city's middle class and the lesser landowners. And some landed dynasties transcend classification, like the Dovara in Cremona, who had to reconcile their "urban" branch of the family with their power base on the important estate on the banks of the river Adda.

Generally, the poorer members of society could not afford to arm themselves, and would not participate in conflicts (however, here too there were exceptions: "Sarjeants" were servants who were employed by great lords and trained in the use of arms, often recruited from the upper-working-class). In the largest cities, like Venice or Milan, there were also systems in place to recruit 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 were given the responsibility to carry out the will of the city council (fundamentally assigning them the responsibility of an executive) but with this privilege were expected to provide arms and armor for all those in their neighborhood in time of need.

So in situations where the Milanese were bringing out their full armed citizenry, the Capitani would have armed the citizenry with pole-arms. If time allowed, their sarjeans would have drilled some discipline into the recruited citizens, however they might just as easily resort to pushing against the battle-line in order to jam draftees together and make sure they presented a tight line of spears while being unable to turn and run. Middle-class citizens, mainly artisans, would generally be expected to own a crossbow, while the upper-middle-class, like doctors and lawyers, could be expected to own a good set of arms and armor as well as a horse, generally comparable to those used by the upper-class. However, because the upper-class consisted primarily of individuals living off the revenue from their land, they would not only be expected to own a good suit of arms and armor but also have had a lot of time to practice with them. In addition, landowners could be expected to own multiple horses and bring more attendants with them; a major advantage when on campaign. The most powerful Milanese landed magnates, like the Torriani, Caronia, Busto, or Baggio, could even be expected to parcel out portions of their estates to trusted relatives and associates, meaning that they could call on an extensive well-armed retinue in times of need.

But by the 14th century, the practice of hiring sarjeans was replaced by the hiring of veterans of conflicts in other parts of Europe, the prelude of the "Mercenary" system of the Renaissance (although I am not a fan of both of those terms).

I answered a similar question on medieval Italian army composition here, and talked about Venetian conflict management in the "Dark Ages" in this other answer, while also talking about medieval mercenaries in this answer. However, there is certainly much more room for discussion and I am happy to answer any additional questions.