As a typical American I know next to nothing about the cultural differences between coastal asian countries. I'm probably a bit racist, but I can't see the difference between Cantonese and Kana. They just look like hieroglyphics to me.
It seems most modern day European and American countries share a common background based in Latin (after the dust settled anyway), but I don't even know what system the current Asian languages are based on. Can someone give me a Cliff notes history of the (coastal) Asian continent?
It would help a lot if you could answer a few questions:
Which countries do you mean by "coastal Asia?' A majority of Asian countries have a coastline.
Are you primarily interested in the history, the languages, or the culture?
As was said, you might want to clarify what you mean by "coastal SE Asia". That being said, the population history of Southeast Asia (in an expanded sense which includes southern China) is often conceived of in terms of "pulses" of population dispersals that left marks that are still visible on the extremely complex ethnography of the region. The earliest inhabitants are often assumed to be the somewhat unfortunately named "Negrito" peoples, who today survive as small hunter gatherer communities in marginal territories of the Andaman Islands, the Philippines, Thailand and Malaysia. These are often seen as the most marginal and "primitive" populations, and thus also seen as the oldest, which isn't the most airtight reasoning but has some genetic support.
The next wave is generally thought to be the Austoasiatic speakers, such as the modern Khmer of Cambodia and Vietnamese. The spread of Austroasiatic speakers is somewhat controversial, but it may be the result of the initial dispersal of agricultural population in the region. Regardless of the circumstances of its spread, it is generally seen as "indigenous" to mainlands SE Asia.
At around the same time is the spread of Austronesian languages. The Austronesian dispersal is actually quite wonderfully well studied, and although many details are controversial there is a very large body of data to each of the opinions. The Austronesian languages originate in Taiwan and coastal Fujian, and sems to have rapidly spready along the island world of Asia, reaching Madagascar in the West and Hawaii in the East (the Polynesians are Austronesian speakers) and including Indonesia, the Philippines and Malaysia. With some very major exceptions (such as the Cham people of Vietnam) the Austronesians do not have as significant a presence in mainland SE Asia, but they have a somewhat outsized role in the historiography of the region.
Now, starting around 500 BCE we can actually trace the spread of peoples and languages through the use of Chinese and later native historical sources. What happens next is that, in large part as a result of the expansion of Chinese imperial power, Tai (in the East--modern Thailand and Laos) and Sino-Tibetan (in the West--modern Myanmar) populations were pushed south from what is now southern China into southeast Asia.
It is worth noting that a language is not a population, although the spread and distribution of a language is often the result of historical circumstances. It is also worth noting that none of these groups were united and should not be thought of as single entities or single historical events--the Shan and the Thai, for example, are very different ethnic groups with very different histories, but are both speakers of Tai languages. But this can be a very general guideline to understanding the exceptionally complex ethnographic map of southeast Asia.
What you need to keep in mind is that the writing system that is/was previously in use for languages like Korean, Japanese, Mandarin, Cantonese, Zhuang, Vietnamese (and others) is really just a writing system, and it's not at all reflective of those languages being related to each other (though Mandarin and Cantonese obviously are). The system of Chinese characters was adopted and in many cases modified to accommodate a wide range of unrelated languages.
The reason for this is that the various states/kingdoms/empires we lump together as China have traditionally been the cultural centres of the region for a very long time. Thus if you wanted to conduct trade or official business with China – as many people did – you were exposed to the system and its use was encouraged. For example, there are Mandarin textbooks that go back to the 15th century which were used to teach Koreans the language and writing system.
So a once-common writing system in the case of Asia is actually not having anything to do with cultural origins, but is instead the result dealing with China over the centuries and the cultural influences that were part of that.
/u/tiako has already made reference to some of the history of this as well in his edit to this comment.