or any other merchant republic, would I have to be a trader? Would I have to have experience sailing to foreign lands and speak many languages?
Concerning Venice in particular, you probably wouldn't have needed to do any trading yourself. The Venetian government, and I highly recommend Madden's book "Venice" for further information, was structured to favour business and commerce, never directly taxing income. In fact, Venice was founded in a different area, called the Lido, in the early 400s AD after the Visigoth invasions, but soon moved to what is called the Rialto. The reason was that as early on as 800 AD, trade had become a major lifeblood on the Rialto, the centre of Venice and today an area filled with American and Asian tourists. Because of the city's foundation as a trading hub, the Venetians were necessarily fond of trade, leading to the aforementioned government laxity concerning taxation on commerce.
Most Venetians would, rather than going abroad to sell their wares, would engage in many sorts of indirect trade. For example, the "colleganza" was a major mean of trade for commoners. This process would mean that a community would pool money to invest in a ship going abroad without placing any actual goods on it. This benefited merchants, since they now had capital, and those who invested, since they were promised and very assuredly received interest off this loan upon the ship's successful arrival.
Another means of investment was in the ship directly, either using the ship as a transport for your own goods or buying space on it to be filled by the merchant who owned the ship. The merchants, in the first case, would return the interest and the price of the space in exchange for commission. In the second case, they would return interest on the price of the space (again, just a source of capital) and keep the profits from the goods they placed inside it.
Furthermore, it has been shown that virtually every Venetian citizen during this era had a bank account, and most Venetian banks were international. This meant that funds in any amount could be transferred with only pieces of paper across the entire span of Venice's sphere of influence. Venetians could, in this way, buy things from abroad without leaving. Moneychangers often sat in St. Mark's Square where the pinnacle of the winged lion and St. Theodorus now stand, eager to excahnge currencies and to facilitate the transfer of cash between accounts in banks with which they were familiar.
Therefore, it was usually trained merchants alone who performed the actual trading and transactions in distant lands. However, most Venetians did engage in trade indirectly and reaped profits thereof.