I chanced upon an image of Ptolemy's world map and noticed that even though the dimensions are wrong, the map does mention China, Sri Lanka and India.
My question is how much and subsequently how many ancient Indians knew about the Romans, and how much and how many Romans knew about the Chinese and the Indians and all the other various kingdoms bordering the Indian ocean?
Also, did other people like those in Somalia and Assyria knew about the Roman empire, and did they communicate with each other?
(Ps: I would also be grateful if you could direct me towards other sources. Thanks a lot!)
I wrote an answer to this question here (pertaining specifically to philosophy):
(And I suggest some things to read too!)
You might be interested in this post I wrote on the Greco-Roman world and India: https://www.reddit.com/r/AskHistorians/comments/gvxegt/thomas_the_apostle_went_to_preach_the_gospels_in/
I cannot provide much information on China unfortunately, or no more than you could find Googling yourself. There was some trade via the Silk Road but I think it was indirect and as far as I know there was little contact between the Roman and Chinese states.
For India there is a little more, which makes sense as they are geographically closer areas. Some ancient Greeks were aware of India. For example, Herodotus refers to India in his Histories (3.97-106) and Alexander the Great arrived there near the end of this life. Comparing the accounts from Herodotus to someone like Strabo, you can see a clear increase in the amount of knowledge amonst at least some in the Greco-Roman world. It is unlikely that your average Roman citizen had much of a conception of India, but I would have thought most educated citizens would have had some awareness. There are records of embassies being sent from India to the Roman Emperor, starting with Augustus, but diplomatic relations were minimal in terms of their practical worth.
In terms of ancient sources, there are a few you could follow up on. Strabo, Ptolemy, and Eratosthenes would be your "go to" ancient geographers and their work is available free on http://www.perseus.tufts.edu/hopper/.
In terms of modern sources, Romila Thapur's Early India: From Origins to AD 1300 (2003) would be the main work in the area. Curiously, she refers to the fact that although there are numerous references to India in Greco-Roman sources, there is almost no trace in Indian sources. Roman coins have been found on the Malabar coast, so we know that trading took place, but by and large they've left little obvious impact.
Regarding Somalia and Assyria - do you mean Assyria in modern-day northern Iraq? In which case, it was in the Empire at one point. On Somalia, I can't offer much more than my comments on China, but I think it would be closer to the Roman-Indian relations.