Military history is kind of a gray area for me in that I'm not particularly interested in it, but I've inevitably bumped into it enough in order to have written out a few decent answers which I'll link for you below. I'll start with a link that will function as a TL;DR: I answered a similar question about three years ago, which you can read here. Below, I'll try to offer a bit more context while also being concise, and improve upon what I wrote in the link above (and the links below).

In between the fall of the Roman Empire and the unification (itself an armed struggle, and an ultimately successful one at that) in the mid-19th century, the Italian Peninsula was variously governed by two and a half kingdoms, two and a half empires, countless city-states, autocratic monarchies, principalities, and republics of varying shapes, sizes, and institutional qualities. Summarizing how each of these individual polities organized themselves during armed conflicts is going to be difficult, especially because organizations changed over time. You yourself can imagine how Venice in 1222 (during the Fourth Crusade), 1400 (during the Lombard Wars) and 1516 (During the War of the League of Cambrai) might have been a vastly different place.

But we can still reason in terms of general ideas. A theme common in most of Western Europe as the Roman Empire evaporated is the transition from the tax-and-administer structure of the Empire ("Politics of Taxation") to a more localized land-and-privilege structure ("Politics of Land") that consisted of, predictably enough, the ruling elite replacing rights over taxation with rights over land. This replacement occurred to some extent in Italy, but markedly later than in the rest of Europe, only ever appearing in the "Second" kingdom of Italy in the sixth century, rather than in the "First" kingdom of Italy whose emergence marks the definitive end of the Roman Empire. Indeed, in the First Kingdom and the subsequent Byzantine restoration, painstaking efforts were taken to keep the old Empire's institutions as intact as possible. This was, predictably, unique in Western Europe.

What this means in terms of political organization (and as per your question, military organization) is that Italy never fully transition to the "Land-and-Privilege" system you'd expect to find in the rest of Western Europe. Sure, it's also important to point out that the rest of Europe also adopted various hybrid regimes, with forms of city-based taxation systems remaining intact (well preserving some reduced form of professional soldiery) but the difference is that in Italy the institutional setup remained markedly weighted towards cities and urban institutions, and the scales never really tipped towards the "Politics of Land" as they did elsewhere (even though, as you'll see in the links, a land-based aristocracy does make appearances; but the way they exert their legitimacy and power remains linked to the city and its institutions).

A few years ago, I wrote this short, general answer on the composition of Italian Militias in the Late Middle Ages which might interest you.

You might also be interested in this even more general discussion on Medieval Military Force composition, where I dive into some Italian examples, where I'll emphasize this specific reply on equestrian culture in Medieval Italy.

I also wrote this longer, more specific answer on the composition of military force in Venice in the Early Middle Ages which looks look at the more specific mechanics of armed conflict resolution of an admittedly unique Italian Polity.

I also answered this other questions on a specific and decisive battle for Medieval Italian History: the battle of Legnano (although I'll dive into pedantry here and say I don't really believe any single battle can be "Decisive" within a grander social, political, and economic trends — in fact I'd say these factors are what decide battles more than the other way 'round).

This last, somewhat generic, answer on Italian warfare in the 15th century might also interest you.