Your options: walking, riding a horse or mule, by carriage or wagon, and by boat.
The Romans built a paved highway network of an estimated 50-80,000 miles connecting every province in the Empire. The roads were built primarily to facilitate the movement of the military legions. Commerce benefited as well. It was possible to stay on highways from the west coast of Africa through Egypt, Asia, Greece and on to Italy, Gaul and Spain. After the fall of the empire there was not an equivalent highway system until the 18th century.
Of interest in this story is the fastest mode of travel—by ship. Rome to Alexandria in Egypt could take about seven to ten days under optimum conditions. Ships traveling north to south benefited from the northerly winds that predominate in the Mediterranean world. The return trip, against the prevailing northerlies, could take one to two months. This was an important consideration in Rome’s massive grain imports (the annona) as the grain came from Africa and Egypt, the “breadbasket” for Rome.
Merchant shipping dealt with the same north-south issue, as well as weather issues and lack of modern navigational aids (no compasses or astrolabes). Maps and knowledge of geography, currents, tides defined an experienced seaman. Thus merchants, protecting ship and cargo, were often conservative—sailing along coastlines where safe harbors could be quickly accessed.
All these factors were considerations that our seafaring characters had to deal with.
As a “fun” aside, check out the ORBIS interactive mapping site referred to in the Suggested Reading. Online you can plot a course from city to city and compare the modes of travel and the associated cost. This site helped immeasurably to plot Gaius’ navigation of Mare Nostrum.