Venice participated in significant conflicts over the course of your centuries-long period, just to name a few: the Venetians were significant players in Barbarossa's descent into Italy; the Fourth Crusade made the Venetians the most powerful force in the Aegean and Adriatic seas; the Chioggia War made Venice the dominant naval power in Italy; and Venice's acquisitions on mainland Italy turned the city into a European player. Each of those events, and many more, modified the ways Venetians thought about state-level conflict resolution.

It's also important to point out that large salaried standing armies were not the norm in Italy, or indeed the rest of Europe, after the fall of the Roman Empire. For the most part, Venetians merchants and sailors only really armed themselves to guard against pirates on the high seas, although a segment of the aristocracy certainly imitated the warrior-elites of other states. But on the whole, the Venetians were not aggressive warmongers, although that might also depend on your definition of war. While Venetians did not have to think much about organizing a land army until the fifteenth century, they did invest a lot of time and energy in naval affairs. But rather than large-scale, campaign-style wars, Venetians were much more interested in low-scale conflicts, expending a lot of time and energy in imposing a Venetian-dominated order in Dalmatia, in Istria, in the delta of the Po River, and on the waterways in the Friuli, in order to make landfall safe for their ships and clearing the waters of pirates.

But naval raids were not the only method through which Venice imposed itself on Istria, Dalmatia, the Friuli, and the Po delta; if Venetian merchants were pushing for action against pirates and lawlessness, it was because those regions were integral part of the Venetian macro-economy, and whose merchants and aristocrats had tight economic, familial, and social ties to Venice. As early as 932, the Istrian port of Capodistria had sworn fealty ("federata" or "fidelitas") to the Republic, and by 973 neighboring Isola followed suit. Little over hundred and fifty years later the city of Pola was also persuaded to swear fidelitas to Venice. In the following centuries, Venice managed to secure the submission of nearly every major port along the Dalmatian coast and several islands in the Aegean Sea: in 1220 Cherso, in 1236 Arbe, in 1237 Ragusa, in 1254 Curzola, in 1278 Lesina. Already in 1212, Venice had seized its prize possession: the island of Crete, which they called Candia.

Of course, it's also true that Venice's Thalassocracy would grow thanks to the Fourth Crusade in the 13th century. But Venice's active role in that crusade was something of an anomaly, and even then, the Venetians' political and logistic role far outmatched their military one. In fact, the Venetians' whole strategy was to get the Crusaders to do most of the military work in order to do things that would have been much more difficult for them to do alone. I wrote more about the political perspective of the fourth crusade here.

So given their propensity for diplomatic and naval affairs, at the end of the day what kinds of fighting men did the Venetian government have at its disposal? Certainly, even on naval expeditions the Republic could count on a small corpus of well-armed and richly armored aristocrats holding jobs as naval officers, like ship captains and navigators. In times when the whole state mobilized for war, the wealthiest and most important of these could be expected to be followed by elaborately armed retinues. And it's true that the wealthiest venetian families also maintained rich stables, as indeed specified above; that might seem surprising, but it's important to point out that prior to the city being completely paved over the course of the 16th and 17th centuries horses would have been, if not ubiquitous, certainly not uncommon. Before the city was densely built up it was fundamentally a community of proximate islands and sandbars, with low wooden bridges easily traversable on horseback, and low-density groups of houses divided by fields, orchards, and gardens. When the bells of St. Mark's called the aristocracy to the great council, the locals called the sight of men on horseback making their way to the Ducal palace the "Trotéra," roughly translatable as "The Trot."

A more "average" sailor or a merchant could be expected to own and know how to use a weapons like crossbow. The crossbow, in fact, was a particularly popular middle-class weapon favored by Venetians. In times of war each neighborhood organized a lottery, where one man would be drawn to serve in the navy while his neighbors were tasked with paying for his arms and armor.

However, defense as well as law and order on Venice's colonies was mostly a local responsibility; with the Republic normally undertaking to keep the city supplied (typically by sea) and paying for a small force of administrators and soldiery to support the local aristocracy in running the city. Although initially drawn from locals, over the course of the fourteenth century these companies of soldiery were increasingly recruited from veterans of conflicts going on in the rest of Europe (a particularly interesting case involved a group of hundred years war veterans contracted for garrison duty in Crete).

How effective the armed venetian sailors and aristocrats could be against the armies fielded by other medieval states is difficult to gauge; there's no real historical encounter between Venetians and other Italian states until the 16th century. Venice's ability to project power was intimately tied to the Republic's maritime nature, the Republic was able to supply and arm garrisons on fortified outposts all over the mediterranean, certainly something that few other states could do. While certainly greatly beneficial for Venetian merchants, it's difficult to gauge how this compared to the large military campaigns organized by other medieval states. The Venetians didn't even measure themselves much with other Italian states: the Republic's status as Northern Italy's major entrepôt exerted a sort of gravity on neighboring communities, such that its expansion was largely peaceful. In fact, on the death of the Duke of Milan Gian Galeazzo Visconti in 1402 the cities of Padua, Vicenza, and Verona all agreed to defect to the Republic. You can read more about Venice's landward expansion here.

You might also be interested in this answer about Venice's conflict resolution apparatus in the first centuries of its existence.