I have removed all of the comments below, including a lot of discussion about the History Channel. Please remember the Rules for Ask Historians before posting.

There is no need to "fill the void", if you don't really know the answer and are not ready to back it up. Someone knowledgeable will come along in a little while. It's also explicitly mentioned in our rules lack of previous answers isn't an excuse to not follow the standards. Either someone knowledgeable will answer the question or they won't, your posting a partial answer is unlikely to change that and yes we'd much rather have questions stay unanswered rather than be answered with incomplete or misleading information.

The closest thing would be the postal system or cursus publicus which had frequent stops where imperial officials could change their horses so that they always had a fresh mount. However, this was hardly public despite the name. You needed a diploma from the emperor himself to use it, but those were given out to all sorts of imperial officers, tax collectors, military officers and powerful families as an honor. The Wikipedia article for this is actually pretty rigorous and includes primary references (the warning at the top is mostly because the author relied on personal research instead of quoting from secondary sources).

As for regular transportation, it was usually wrapped up in either trade or war. You could travel by booking passage on a ship bound for your destination (there probably weren't any passenger ships per se) or join a group traveling by foot out of your city. For example, when the wealthy virgin Melania decided to visit the monks in Egypt and Jerusalem she took a ship from Rome with a whole posse. When she traveled between Jerusalem and Constantinople, her posse hit the road and made the trip in record time probably by using the imperial post (she had connections to the imperial household).

What about the people who couldn't afford to pay a posse or hire a ship? There is not much evidence for how these people traveled, but analogy to later periods can help. In early modern England, people met at taverns near the town limits to gather into a group for travel. Basically you would have the tavern for people going north, a tavern for people going south, etc. I imagine that people in ancient Rome did something similar. They met at a common location that was known as a place where you can join a group for traveling and departed together as a means of protection.

I've studied ancient Rome for a while and can't think of anything that completely fits the bill. Nor would I really expect one. Peasants would probably rarely go far from their home towns, Rome was walkable, and the people who really need to get places would either be taken there by the state if they were soldiers, or would be rich enough to afford travel by horse, cart or sea privately.

A few things do come back to memory though.

The Romans did have a system of horse relays (i.e. swap your tired horse at a special horse swapping station for a fresh one; a really good ancient system of travel) called the cursus publicus but it was for agents of state really, carrying military or state messages to where they need to go. So that doesn't qualify, quite. Anyway, if you're interested in it Suetonius says:

To enable what was going on in each of the provinces to be reported and known more speedily and promptly, he at first stationed young men at short intervals along the military roads, and afterwards post-chaises. The latter has seemed the more convenient arrangement, since the same men who bring the dispatches from any place can, if occasion demands, be questioned as well. (Suetonius, Aug. 49.3)

Perhaps slightly better for your purposes is the fact that the Edict of Diocletian's edict on maximum prices included some bits about travel.

This edict was enacted by Diocletian in 301 AD in an attempt to halt the rampant inflation that was happening at the time (through debased coinage I think). It set the maximum prices people could charge you for things Now, I don't know if you know anything about economics but this is always a terrible idea - forcing people to ask for less money than their goods and services are now worth. It was mostly ignored.

But it did set the maximum price for both sea and land (oxen-drawn) freight, so that probably constitutes 'standardised fees' at least. source.

Sorry I couldn't help more. It's a great question!

EDIT: Typos. Also I forgot to mention the name of the horse post system.