Selling and reselling ships was a normal occurrence. Unfortunately I can not say if the ships would be always commonly available in a port, as that probably depended on the situation and supply and demand at the moment, but we have numerous examples of "used" ships being sold. One example I have at hand, but which just goes outside of your time frame, are the five ships Spanish crown bought for the purposes of Magellan's expedition in 1519. All five were previously used, and had to be subjected to a complete refit. As we happen to have preserved documents of the cost of documents, like this overview from here, and a complete itemized list here.

As we can see from above lists, the cost of ship itself was huge, and even that was dwarfed with the necessary refit of the ship (cost of ships was 1/3 of total cost for refit), and then the costs of that was equal to the cost of preparing a single expedition: provisions, crew wages, merchandise for sale. True, Magellan's expedition was special case, and it was prepared to last really long (few years), and a had a accented military nature of an expedition to unknown, which ordinary merchant's wouldn't do and their cost would be much less. But still it shows us some reference.

This brings me to the second part of your question, of what would "aspiring merchant" do if he wanted to go trade. Well, to be honest, buying own vessel of trade, new or used, was possible but very rare as it would exceedingly expensive. The merchant would need to have exceedingly high capital to do so, at least for larger ships necessary for longer voyages. Instead, if you were a newcomer to trading business you would simply try to find a ship already going where you planned to go and rent cargo space by paying freight charges and send your merchandise that way. If necessary for you to go, if for example there was no one you trusted to handle your merchandise on the destination , you would also have to pay passenger place for yourself to accompany your cargo.

If for any reason you didn't want to do it this way or it wasn't an option, you could charter a whole ship for your purposes. Basically rent the ship (and the crew) from a shipowner just for the voyage. You would have to pay for the ship and for the crew wages and provisions entirely by yourself. As this again would be enormous cost, there is a high chance you wouldn't do it yourself. You would usually find alike minded merchants and form a partnership with them, a "company" as it was sometimes called, where you would pool money and merchandise for the voyage. If none of the merchants owns a ship, you could also find a shipowner either rent a ship, or even include him in the partnership where he gives the ship as his part of investment capital. There were lots of different ways this could be arranged. Example that I am familiar with was the arrangement Henry the Navigator was arranging to merchants going to Africa in the 1450s was that he gives the ship and crew, and the merchant gives the merchandise and they share profits 50%-50% (the other option being that merchant acquires a ship himself, in which case he had to give 25% in name of taxes).

If you would have enough money to actually buy a ship, you might do so. However it was more common for merchants to buy "parts" of the ship instead. For example instead of you buying one ship and owning it 100%, you would find a partner who wanted to buy the smae and buy two ships with each owning 50% of each. In this case, you diversify and mitigate risk, so if a ship is lost, or profits of a voyage bad for any other reason, you don't lose everything but only half. Sure, there were rich merchants single owning ships (and multiple ships) and either chartering them out or using for their own expeditions. There was also men rising through ranks of mariners who if they were lucky earned enough to buy a ship, and they often bought the ship in entirety. But merchant men, aspiring and not yet super rich, if they did venture to enter ship-owning, preferred to form such partnership and shared ownerships of vessels.