First off the Greco-Roman world extends over a long time period. Depending on how you measure it that could span a millennium. So the answer to your question changes depending on when you are travelling, (and the season of the year; some seasons you travel by sea, and some you must travel by land) and where you are travelling.
Beyond that, the answer depends on your social class. If you are a plebian, you're going to walk. If you are a slave, and IF you are permitted to travel, you're going to walk. If you are of the Senatorial class, you may ride, or you may be carried. The equestrian class were originally those who were wealthy enough to own/support a horse and armor, and notionally if you were below that threshold, you were not going to ride.
While I cannot prove it, I would be very skeptical of any claim about public transport. Public services are (in general) an artifact of the modern welfare state. Rome provided some public services (the Urban Praetor was responsible for the sewers), but generally those services were provided grudgingly and poorly. There is no fundamental public interest in speeding travel.
I suspect that Romans walked, with very few exceptions.
The following paper models potential transport, and includes speed estimates.
Travel in the Greco-Roman World
Most of the travelers along Roman highways were caravans, camels, horses, and donkeys.
Travel & Transportion in Ancient Greece is a bit older than you asked but does discuss carts and chariots.