I'll confine myself to a few comments on overland trade between the Roman Empire and Han Dynasty China.
The Romans had two trade links with the Far East. One was a maritime route that began in Roman Egypt, and ran along the Persian Gulf and across the Arabian Sea to the west coast of India and Sri Lanka. The other was the famous overland route that connected the oases of the Tarim Basin (in Xinjiang) with Antioch and the other trading centers of the Roman Levant.
At least once, Roman merchants reached China via the Indian route; in the reign of Marcus Aurelius a group of either shipwrecked or very adventurous Romans appeared in the Han emperor's court at Luoyang, and offered the (unimpressed) emperor "elephant tusks, rhinoceros horn, and turtle shell" (Hou Hanshu, section 12). More commonly, silk was traded to Indian princes by Chinese or central Asian merchants.
[EDIT: A great deal of silk reached Rome via the Indian Ocean route, but u/Celebreth has an excellent response that covers that trade network. My answer is confined to the overland routes.]
No single merchant carried silk the entire way from China to Rome; that simply wasn't time- or cost-effective. Instead, merchants shuttled goods from one city to the next, probably seldom going more than a few hundred miles. Luxury goods like silk, of course, would tend to be shuttled farther, particularly since the Roman market was known to be so lucrative; but even silk would stop along the way, particularly in bad weather and times of unrest.
I know of no passage that actually describes how long the overland journey would take from China to Rome. I would guess, however, that it took most bolts of silk nearly a year to make the trip. Let's say our silk shipment left Luoyang in the spring of 100 CE. Since the Tarim Basin had been brought under firm Chinese control in the mid-first century, the journey would have been fairly painless up to about the modern city of Kashgar - perhaps only two or three months. Then the silk would be passed on to merchants who operated beyond the imperial frontiers.
The cities of the region just to the west (ancient Bactria) were probably under the control of the Kushan Empire. The first language of many of the merchants, however, was probably Greek - a legacy of the central Asian Hellenistic Kingdom of Bactria. From this point onward, the silk was probably loaded onto camels. Bactrian camels can travel great distances at surprising speeds. But most caravans probably managed only 10-15 miles a day, largely because they would have stopped at any possible place along the way where a profit could be made. So let's assume another month or so to cross Kushan territory.
Somewhere in modern Turkmenistan or Afghanistan, the caravan would have crossed into Parthian territory, and the silk was probably passed along to another batch of merchants. The Parthians had inherited the network of roads first created centuries before by the Achaemenids, and the silk (probably still on camel) would have moved fairly quickly- let's say only three or four months to make it the 1,200 or so miles to the western frontier.
At one of the great caravan cities on the frontier - probably Hatra in Iraq or Palmyra in Syria, the silk would have finally crossed into Roman territory. It would then have moved quickly (say a week or two) to one of the major ports: Antioch/ Seleucia in Syria, Caesarea in Palestine, or one of the smaller harbors between. Depending on the time of year, the journey to Rome (or rather, Rome's port at Ostia) could have taken only a few weeks. If the silk arrived after the close of the sailing season (and it probably would, if it left China in the spring), it would take as many months.
So silk from China might be able to make it to Rome in eight months or so - but I suspect that a year was more common.
Aight, so this question is right up my wheelhouse, and it's a legitimately good one. The Romans loved silk. After Augustus opened trade up to the East as he did, its popularity - along with other Eastern imports - skyrocketed, and demand ensured a thriving trade with the East.
There are two major answers to this question, one of which I think is what you're asking, and one of which is a more honest answer. I'll see if I can adequately cover both :)
First of all, the vast majority of Eastern trade was conducted through seagoing merchants. The merchants would depart from ports on the Red Sea in Egypt (oftentimes Myos Hormos or Berenike) in July to take advantage of the monsoon winds. The exact route they took depended on the time period: as trade continued and increased, Roman merchants learned more and more about the geography of India, realizing that it was possible to sail directly across the ocean to points in India, rather than coast hopping. As a result, they built up their ships to be able to handle such voyages and were able to cut the travel time from their Egyptian port to an Indian port 5000 km away to about 70 days - 30-40 of which were extremely hard sailing over open ocean with almost constant heavy winds. Once arriving in India, sometime in early-mid September, a trader would be able to spend 6-10 weeks before having to make the return trip, which was possible from early November to December. This is where the silk comes in: India was a thriving market for silk, and by far the largest exporter of silk to Rome. Many of their silk products came from their trade with China: overland, by river, and by eastern sea travel. Barygaza happened to be one of the largest of these, but overall, it's difficult to estimate the travel time between insert Chinese source and Barygaza itself. For the Roman merchants though, it's almost clockwork. The ships would take another 2-3 months to make it back to Egypt, resulting in a total travel time for this merchant of about 200 days.
But THEN we're looking at that piece (well, probably a bundle tbh - these trade freighters dealt in bulk) making it from Egypt to Rome. So the silk had to travel from Myos Hormos (for ease of working) to the market at Coptos, where it would have been processed, taxed, and sent down the Nile to Alexandria. At Alexandria, the shipment would be processed again to ensure that all proper taxes were paid, then sent to Rome. Depending on the route that the shipment took - and remember, this would be in February or so, which is not the optimal travel period for the Mediterranean. A cautious merchant might wait for the gentler spring weather, but let's pretend we want to travel now, for ease of not estimating "waiting." There were two primary routes from Alexandria to Rome: One around Sicily and one passing through Carthage. Both places would be great stopping points for a merchant to offload some goods and to resupply, which might take a day or two in itself, and then travel onward. All in all, this trip would take between 45 and 60 days.
So overall, by sea, a Roman merchant would spend ~250ish days to get silk to Rome. If you want the breakdown of that, read the above.
By land is a whole other story. Throughout this, I want to emphasize two things:
Land trade with the East existed, happened, was constant, and was quite prosperous.
Land trade with the East was far less than the sea trade, records of it are far less thorough, and we don't have a Periplus of the Erythrian Sea for the land trade. Chances are, your Roman silk was brought from an ship that went to India.
Okay, disclaimers outta the way. I'm gonna go ahead and pillage from one of my old answers about Roman caravans to discuss this a little bit. We know about routes between Parthia and China (Chinese envoys were generally received with high aplomb and they lavished gifts on each other), and we have a few trade cities that sprung up on the eastern frontiers of the Roman Empire (and a bit beyond, really, which is potentially why they were so prosperous). You may have heard of these names a couple of times: Petra and Palmyra. Petra is famous because of its glorious, extant, architecture, even if most of it is closed to tourists for the purpose of preservation (see: tourists who vandalize ancient sites, like carving their names into the Colosseum or stealing pieces of Pompeii). Palmyra is famous for a much sadder reason: the ancient site recently suffered enormous damage at the hands of IS. Both of these sites were trade cities. Petra was a "Last gas station for the next hundred miles" kinda place - it was built around an oasis and water was one of its most precious resources. As a result, it became the natural gathering point for merchants heading both east and west, and its prosperity fueled the Nabatean Kingdom for centuries. Goods came in, merchants were taxed appropriately, merchants bought the supplies that they would need for their journey, and merchants sold goods to those who wanted them. It was a busy and prosperous place and, if there was a Silk Road between Rome and China (again, unknown because we have no direct evidence for one), this would be the city where it started and ended.
What is more likely, however, is the idea of periodic trade. For example, the Periplus of the Erythrian Sea records that:
There is a very great inland State called Thina (China). From there, silk floss, yarn, and cloth are transported by land via Bactria to Barygaza and via the Ganges River to the Malabar Coast.
From there, Roman merchants avidly traded for these resources, selling them for a huge markup back in Rome. Another example of periodic trade is the idea that Chinese merchants would sell goods to nomads, who would trade these goods among each other across the steppe until they reached Rome through a more northern town - Palmyra. There are no extant documents about the silk trade in Palmyra, though there's plenty about trade in general, but there's a huge amount of physical evidence (literally thousands of fragments of silk and other foreign textiles) that has been found at the site. Palmyra, like Petra, was a caravan city. It was founded around an oasis, and was the last port of call for merchants heading East - this time, more heavily focused directly toward Parthia. Like Petra, Palmyra was founded independently, but unlike Petra, it was conquered much earlier, and therefore gives an example of a city directly under Roman auspices. And it seems like that overhead didn't hurt the city at all - in fact, Palmyra just continued to grow and develop, with its apex sometime in the second century.
u/PrettySlattern had a great response to this question here: https://www.reddit.com/r/AskHistorians/comments/1dzsbn/how_long_time_did_it_take_for_a_piece_of_silk_to/
