I guess I'd compare this to how a western civ course would cover the renaissance, napoleon, the enlightenment, and the world wars. What events in Asia are like these in terms of dominance.
Speaking for a China-centric standpoint, but also the aspects of Chinese history the west is most familiar with:
Classical Chinese philosophy, the essentials that you've heard about in the west but that are equally important throughout asia being Confucius, Lao Tzu, and Sun Tzu.
Qin unification of China. The first emperor of Qin is of course famous for the terracotta warriors and the Great Wall. The Han Dynasty that came afterward, well you just need to know that it carried on much of the Qin ways but with less brutality, setting forward the example for Chinese empires the next 2,000 years.
The fall of the Han dynasty (an event that has its similarity with the fall of Rome) and the Three Kingdoms, a legendary period of heroes and villains, empires lost and gained. It is ingrained in east asian culture beyond just china through the famous novel Romance of the Three Kingdoms.
Tang Dynasty, its poetry and culture also influential across east asia, and the dynasty arguably the cultural genesis of much Japanese culture.
Mongol invasions. The clusterfuck that affected everybody everywhere in Eurasia.
Early modern interactions between China and the West, stretching the period between Matteo Ricci and the Macartney embassy.
The 19th century disintegration of Chinese state power from both external and internal conflict. Opium Wars, Taiping Rebellion, Boxer Rebellion, First Sino-Japanese war.
End of the Chinese empire and the beginning of the warlord period, which coincided with the fight between nationalists and communists.
WW2 in China. Nanjing massacre.
Communist political military victory after WW2
Communist tribulations: Great Leap Forward, Great Famine, Cultural Revolution
Capitalist turnaround: Deng Xiaoping, Tiananmen, becoming the world's factory.
Apols for the euro-centric bias, but seeing as you're taking an Asian history course for the first time, from what I presume is a western perspective, these would likely be the essential points from that perspective.
Is it possible to narrow this down any? Asian history is such a huge topic. Is there a particular time-frame you'll be studying? Will you be focusing on Asia as a whole or a particular area? I would distinguish
Central Asia; Turkmenistan, Kazakhstan area
Western Asia; roughly the Middle East and the Near East including Eastern Russia
East Asia; China and Japan area
and South Asia; India area.
If you're looking for a general introductory overview I would start with reading about the Yellow River and the Ganges River as cradles of Chinese and Indian civilisations respectively (the obvious parallel here is with the Nile in Egypt.) this might seem a bit esoteric but they are so significant in the foundational stories of these two great Asian powers and they will lead you to many intriguing jumping off points.