As for India, sparse relations with Europe go back to the 5th century BCE - just think of the Persian Empire, then Alexander, the Greek edicts of Ashoka etc. The Romans of course knew of the existence of India as a distant land in Asia, but regular trade relations began with Augustus. We have written records of Indian envoys at Trajan's time, as well as Greek peripli describing the routes around the subcontinent. More info here.

The Periplus of the Erythraean Sea describes, beside India, many places on the East Coast of Africa, including realms in modern-day Ethiopia and Somalia. Apparently these East African kingdoms had some notions of Greek culture and literature, probably acquired through Ptolemaic Egypt. Rome would have been known at least as the origin of the many European traders who followed this route.

Thanks to written sources from both parts we know that Han China and the Roman Empire had trade relations, though they knew very little of each other... se here for a more detailed account. Basically, Rome knew of the distant land of the Seres, which produced the precious silk; China knew there was a sort of counterpart of China in the west.

With Northen Europe we have less information. Roman geographers and writers knew of Ireland (Hibernia) and of Scandinavia (Scania, the tip of Sweden, believed an island). Trade with these countries should have existed, as we find some Roman artifacts there. Aside trade (which could have been conducted through a third), we don't know of Roman missions of the sort. It has been proposed that Juvenal, Satires II, 159-60 spoke of a Roman mission to Ireland (Iuverna), but it is controversial. More info here for Ireland, and here for the Germanic peoples.

We may add to the knowledge of Rome through trade and (scanty) embassies, the spread of Christianity. The Roman Empire was not only the place where it spread originally, and had its most important centers (not just Rome: Jerusalem, Antiochia, Alexandria); it was also the background of the story of Jesus, from Augustus' census to Pilatus judgement. So when from Alexandria Chistianity spread southward up to Ethiopia in the 3rd-4th century, we can say that Ethiopia knew of Rome, at least in a limited way.

Check out my answer to a similar question. Essentially Hellenism became a cultural language for the elites who were in any way tied into the Mediterranean trade network.

Well at the Empire's height most of North Africa was Roman territory so they certainly knew, and that knowledge would've extended further south to whatever kingdoms and peoples Roman Africa traded with. There was trade with India and though they never directly met, both China and Rome were vaguely aware of each other as large powerful states through middleman countries like Kush and Parthia. The Han thought of Rome to be a sort of counterpart to China that they called Daquin and both sides did try to make direct contact multiple times throughout history, although nothing solid was ever reached.

Ireland certainly had contact with Rome and traded with Roman Britain, while I can't say for Scandinavia. I'd assume there would be some basic knowledge of Rome in the north, and the Romans knew that the region north of Germany was peopled by 'barbarians' who came south time to time, but Germania was Rome's limits in Europe and to my understanding was something of a most isolating Rome from the far north.