I am not sure what you mean by "Republican period" but I can give an answer for 15th around the "peak" of Venetian naval strength.

First to briefly describe the Venetian 'navy'. It consisted of state owned galleys (few if any sail ships were such state-owned) of usually two types: the merchant 'great' galleys and war 'light' galleys. The merchant galleys were used in Venetian convoy system for trading valuable goods to/from particular locations (Alexandria, Beirut, Flanders, Constantinople, etc) and in this system the galleys were "auctioned" for a particular voyage to a highest bidder. The person - or usually a partnership of several people - who won the auction could select whoever he wanted as a commander - called a patron - and he had to be a noble. War galleys - which served either as Guard - patrolling certain seas and locations - or stored on land and deployed when needed fir action, would have the commander - in this case called sopracomito - elected for one season directly by the State. To be eligible for election you had to be a noble, over thirty years old and usually of good standing and in particular good financial status as often they had to pay advances and other costs for crew and provisions from their own pocket (which could maybe be reimbursed later, but definitely financial liquidity was needed. These position was highly coveted and many noblemen did their best to secure a sopracomitoposition for themselves.

Now, this command post (patron / sopracomito) was basically the only officer rank on board that was noble. All the rest of the officer ranks were non-noble, and basically the ship was ran by non-nobles with the patron/sopracomito serving as overseer for trade/war matters. Now, it is not unreasonable to assume that in order to become a commander you had to have relevant experience. But how to get relevant experience, if no other officer posts were available for nobles? Well, one part of the solution was that the commander could bring aboard his retinue, like pages, servants, or just fellow passengers, and this was regularly used for commanders to 'accustom to the sea' their children or relatives or closely connected people of trust. Strictly speaking these people brought as retinue were not "in the Navy". They had no authority on the ship nor were they paid by the State. They were solely on the upkeep of the commander who had to pay all their expenses (or whatever other arrangement was made, but definitely not on official rolls).
The other option to serve as a noble on board was to be elected to be a 'bowman of the quarterdeck'. As opposed for ordinary bowmen who were non-noble, who counted 20-30 on board and were positioned on the benches with the rowers, the bowmen of the quarterdeck were noble and were stationed with the commander. There were initially four of them, then six, then eight for each galley. Their salaries and supplies were provided by the State/galley as part of the crew (they were naturally paid more). The secondary literature I am drawing from openly describes these positions as intended by the State to give young, poor, experiencing-hardship nobles a chance to get experience on the seas and opportunity to earn some money trading. They didn't have any offical say in running of the ship and were more intended as soldiers. However if a commander would die on duty, the next commander would come from the nobles on board. Initially the age minimum for bowmen was twenty, but it was lowered to eighteen, and exceptions were frequently granted for younger.

So how young the Venetians were when they started 'going to the sea'? Hard to say. One commander recorded he started taking his son along with him when the child was only four. Another noble records he went on his first voyage at fourteen under his uncle. But it's unclear if he went as a page/retinue or a paid bowmen. In each case that same noble was recorded to be going as bowmen on voyages at around twenty to twenty-two years old. From this few cases we might paint the picture that Venetians nobles started 'serving' (or being employed) on vessels at ages 14-20, with some frequently experiencing ship life and participating on voyages even younger if their parents/relatives would take them.

As a final note, I would just reference the non-noble ranks on board. The highest non-noble rank on board was a comito - usually translated as master. For all intents and purposes he was the main person in charge of ship for all matters naval. Although sopracomito could always overrule him. Second in charge was a paron who was also in charge of the bow (front) of the ship. They were aided by several men called nochiero who were lesser officers, and first step above oarsmen (where there would be junior and senior ones). All these officers could (and would) come from the oarsmen ranks and rise through ability. However all the officer positions on a galley were given through elections in Venice (and you needed to be a Venetian citizen, which could be gotten around by marrying a Venetian woman).

It was deemed beneficial to know how to read and write for junior officers, and for higher ranks you needed knowledge of mathematics and navigation. How young would you start? Hard to say, but for example Venetian conscription laws for oarsmen in time of war recruited from men aged twenty to sixty. As oarsmen were voluntary and often in short supply, younger men were often excepted. So again we may consider age of start being 16-20 (maybe as low as 14) if you were built well enough for the hard job of rowing.