The most far away place that we know the Romans knew of was China, which is also the furthest any diplomatic mission went.
In the Book of the Later Han a Roman embassy is recorded as reaching Emperor Huan of the Han dynasty in 166 AD. It's possible that this was not a proper envoy but merchants passing on information, as the Book of the Later Han inflates the importance of visitors to bolster the Huan's prestige (or so I've read, I am no expert on Chinese history myself). Another mission, this time definitely a formal Roman envoy, reached Emperor Wu of Jin and is recorded in two sources, the Book of Jin and the Wenxian Tongkao.
Roman knowledge of China came mainly through merchants. We know Roman merchants often made their way to the South China Sea from literary sources and from archaeology. Sources from the 1st century BC (Virgil, Horace, and Strabo) make only vague references to the far east as the place where silks come from. They do not display much understanding of China as a political entity. That being said, the Roman elite of the time were quite interested in these people in the far east, as Strabo briefly references a debate between writers on the lifespan of the "Seres", as the people were generally called. The Seres were thought to exist between the Indians and the Scythians at this time.
In the 1st century AD the Romans seem to have developed a more nuanced understanding of the far east and a distinction between Serica, Serve, and Sina appears. Serica was to the north, Serve to the west, and Sina to the south, all according to Pliny the Elder. The name Serica comes from the word for silk (sericum in Latin), but Sina appears to be derived from a latinisation of "Qin". At times it seems the two were interchangeable with other attempted latinisations. It is also unclear what is meant by Sinae, which could refer to the territories or a broad ethnicity. Different latinisations follow different declensions of Latin, so it's all a bit of a mess.
By the 2nd century AD the world atlas of Ptolemy demonstrates that they understood that there was a place called Sina roughly north of what is now the South China Sea (image of Ptolemy's map here: https://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Ptolemy_Asia_detail.jpg/800px-Ptolemy_Asia_detail.jpg) and another place called Serica to the north of a major river, probably the Yellow River (https://upload.wikimedia.org/wikipedia/commons/2/23/PtolemyWorldMap.jpg). The map comes with the description:
"The inhabited part of our earth is bounded on the east by the Unknown Land which lies along the region occupied by the easternmost nations of Asia Major, the Sinae and the nations of Serice... The eastern extremity of the known earth is limited by the meridian drawn through the metropolis of the Sinae, at a distance from Alexandria of 119.5 degrees, reckoned upon the equator, or about eight equinoctial hours..."
From this it seems they were aware of China, and that's as far east as they got. By Ptolemy's definition of where Serica and Sina were, the Han Dynasty would have controlled both at the time and Ptolemy seems totally unaware of this; it is possible (and this is entirely my own interpretation) that Serica refers to Han China and Sina refers to the broad area of south-east Asia where Thailand, Vietnam, Laos etc. are. Ptolemy also shows an awareness that there is probably something further east, but does not know what it is. This is shown on the map, as the land cuts off just before Korea is meant to show up with "Unexplored Land" written along the right margin.
In order to truly understand the foreign policy and imperial attitudes of the Roman Empire, one must attempt to understand how Romans conceptualized the world, or oikoumene. In antiquity, there was not the wealth of information to draw upon when wondering about the outside world, nor was there any point of reference with which to hypothesize on its nature. The desire to understand and possess concrete information about the outside world is obvious from not only mere curiosity, but military and tactical usefulness as well. Most military action took place beyond Roman boundaries, in what amounted to uncharted territory. In the popular imagination, as written about by Ptolemy, Pliny, and numerous others, the world was an oblong shape with the continents of Europe, Asia, and Africa surrounded by a great sea. Furthermore, the world was divided into three zones, only the middle of which was inhabitable: the hot southern “torrid” zone, the temperate zone, and the frigid northern zone. The Roman Empire was well understood and documented (if not mapped), however, knowledge of what lay beyond the frontiers was informed by theory and legend the further one drifted from the border. It is difficult to imagine what was going through a Roman legionary’s mind when he crossed some distant Asian river, or looked upon the sandy Sahara, or boarded one of Caesar’s ships to cross the English Channel; one could liken it to Neal Armstrong’s stepping out of the Apollo rover. It would be similarly daunting for a commander to try to organize and plan logistically when he had no idea how long he would be gone, what the terrain would be like, and what resources would be available. This fact makes the conquests of Rome (and any conquering civilization of the period) all the more impressive given they faced enemies with such a great advantage in intelligence. This understandably led Roman leaders to desire some knowledge of where they would be going and what to expect when out on campaign.
Geographic documentation in Rome usually took the form of an itinerarium, or the naval equivalent, a periplus. This was a list, or perhaps diagram, which listed the stopping points or landmarks along a route and the distances between them. These ranged in detail and accuracy. It should also be noted that maps were extremely rare in antiquity. Only a few examples have been found from Roman times, and Susan Mattern notes that no author nor historian, “including Caesar, mentions maps in a military context or even, virtually, at all.” The maps we do have, as well as itinerarium and other written documents about foreign lands, shed light on Roman perceptions of the outside oikoumene. One of the most prolific of these sources is Claudius Ptolemy’s Geographies. Ptolemy’s work is exceptional in that he emphasizes little but geographic accuracy. In Geographies, Ptolemy describes how a map ought to be drawn, taking into account the 3-dimensional shape of the earth and striving for as much accuracy as possible; he was disappointed by maps that enlarged Italy and Europe and shrank other regions. Much of his information comes from older sources such as Herodotus, as well as more recent data. His goal was to not produce a map that overemphasized Europe’s size whether intentionally or unintentionally, but an apolitical and geographically precise map, which would include ‘accurate’ depictions of Africa and Asia. The information about these mysterious continents was scarce and required approaching with skepticism, so Ptolemy had to just try his best. The map that can be made from Ptolemy’s writing reveals a critical theme of the Roman mind in 125: despite their might and dominance in the region, the Romans realized that there was an entire world out there that was much larger than just what they knew and dealt with normally. This shows a cultural self-awareness that can be used as a benchmark for cultural intelligence and attainment.
In Asia, Roman control usually stopped at the Euphrates River. The vast stretches to the east were less well-known, and mainly remembered in the public imagination of a nearly deified Alexander the Great, who crossed even the Indus and entered India, which was thought of as an one of the largest and most populous parts in the world. Indian goods made it through trade to the west, as did Chinese goods. Actually useful geographic information about these lands is unreliable at best, as merchants are not the most reliable or knowledgeable about geography. China too was dimly known as some distant land, but details were so scant as to be irrelevant. These far-flung places were distant such that they were of no importance to Roman leadership. Southwestern Asia was better known, unsurprisingly. Coasting expeditions revealed a great deal about the lands abutting the Red Sea and Persian Gulf. The Caucasus area was in and out of Roman control, and there were many Roman ports on the east, west, and southern coasts of the Black Sea. The Caspian Sea was thought in Ptolemy’s time to be an inland bay of the northern ocean. Gallus led an expedition into Arabia that was largely a waste; little of worth was found in that vast sandy peninsula and many succumbed to heat and thirst, and attempts to conquer the region were not made.
Similar to Arabia is most of Africa. Although the Mediterranean coastal region was obviously occupied, it is mostly desert to the south. This unsurprisingly cooled Roman desires to explore; coastal exploration and other one-off expeditions, mostly counter-attacks against desert inhabitants, revealed more heat and more sand. However, expeditions down the eastern part of Africa followed the Nile, and made regular contact with Ethiopians at Meroe. Diogenes, a Greek merchant, claimed to have found the source of the Nile at the “Mountains of the Moon,” deep in sub-Saharan Africa; this account is unreliable however. Although the African continent was not explored thoroughly, the results of the explorations that did occur “were hardly commensurate with the exertions entailed.”
Although Europe was the best known continent for obvious reason, it was still not known perfectly. Germany was never properly explored, nor was Scandinavia, which was thought to be an island. Hibernia, now known as Ireland, was thought to be in the frigid uninhabitable zone north of Britannia, which was in turn parallel to both Spain and France. Russia was unknown, as was most of Eastern Europe beyond Dacia and Bulgaria. The Black Sea was known mostly on the southern, eastern, and western sides. A great source of information related to the Black Sea is the Periplus Ponti Euxini. Written by Arrian circa 130, it told of the various ports along the Black Sea and the distances between them. It was addressed to the Emperor Hadrian, who was interested in the geography of the sea especially in the event that a navy would need to be raised. It says nothing of the shape or nature of the sea, only an itinerary of ports and towns along the coast. It also contained detailed reports of the various tribes of people in the area. The Periplus Ponti Euxini illuminates the things that were considered valuable as geographic reference.
A not useful but oft-cited resource is known as the Tabula Peutingeriana, or Peutinger Map. Numerous anachronisms reveal that it has been edited and revised at various times since its origin, the date of which is therefore unknowable. Based on itineraria, the Peutinger Map emphasizes showing overland routes and the cities and waypoints along them. The map mostly shows Italy and Europe, which are vastly overrepresented to accommodate a wealth of detail, but extends eastward to show India, Sri Lanka, Arabia, and more. The anachronisms and inconsistencies demonstrate its failure as a source on Roman geographical knowledge at the time. In addition, it is far less accurate and far worse of a map than contemporary knowledge should indicate. The Peutinger Map is better understood as an oddity and a piece of art instead of a real demonstration of Roman geographical understanding. It is pieces like the Peutinger Map that reinforce the necessity to be skeptical when it comes to Roman geographic knowledge, which was based on unreliable testimony and could be made for reasons other than trying to be accurate.
The tactical desire for accurate knowledge of the outside world was important to the Romans; however, it was checked by the tendency to embellish and self-aggrandize. This tendency takes form in multiple ways, whether it due to returning explorers and generals wishing to make a name for themselves by exaggerating their discoveries, or geographers exaggerating the size of Roman holdings and truncating the lesser-known lands beyond. The conflict between a desire for geographical accuracy and the impulse to self-aggrandize and marginalize outsiders leaves modern historians with a lot of critical thinking to do in this field. Virtually all peoples outside of the Empire were seen as barbarians, living in marginal environments, eating raw meat and living in cold wastelands. The impulse to draw and describe their lands as small, relegated backwaters as compared to the grandiose and central Mediterranean empire of the Romans. For this reason, historians must carefully compare these conceptualizations of geography with sources like Ptolemy, who are more devoted to accuracy.