One thing that a lot of people, including historians who should know better, get wring about ocean travel is the idea that it requires a certain "technology level", that only when you select the "carvel planking" trait can you unlock the "cross oceans" event. Small fishing boats set adrift by storms have crossed oceans, and from a more systematic level the most spectacular overseas voyages of the premodern era were done by Polynesian craft with none of the technological attributes "neccesary" to cross oceans. By and large, the safest answer to the question "how did x cross the oceans with none of the technological developments neccesary to do so" is that such developments were not, in fact, neccesary. This is not to say that developments in shipbuilding technology cannot make voyaging more reliable or comfortable, but we should be careful with concepts of technological necessity.

More specifically regarding Mediterranean ships, while the trireme and others of its style are the most iconic ships of the era, they were actually very specifically war ships, their shallow hauls and banks of oars allowing them the sort of maneuverability neccesary in battle. The actual merchant ships like the ones that followed the monsoon winds from the Red Sea to the coast of India would look more like this, wide hauled barges with deep drafts and large holds, powered by sails rather than oars (the wind is cheaper than labor). These ships could be very large, of a size and tonnage easily equal to those used in early modern Europe, and were capable of reliably making difficult overseas voyages.

If you are interested in ancient shipbuilding there is a variety of different books, but I think Lionel Casson's Ships and Seamanship in the Ancient World, while quite out of date now, is still an excellent introduction.