I'd particularly be interested in seeing a depiction of warfare, or daily life under an empire or something.
I have images of this stuff when I think about Medieval Europe, or Ancient Rome , but this stuff draws a lot of blanks in other parts of the world.
Any suggestions?
There's tons of Chinese stuff like that, but I haven't seen enough to recommend. I did watch Three Kingdoms (2010) and found it entertaining and accessible (though it is 95 episodes, so prepare to dig in), highest budget Chinese show when it came out. It takes some creative liberties I think so you could also consider the 1994 adaptation, don't know anything about it but it looks like it got good reviews. If you don't have time for those, there's Battle of Red Cliff which is a movie about a famous battle from the era.
For less war and more daily life type stuff, you could look for adaptations of Dream of the Red Chamber. Honestly just look up adaptations of any classic Chinese novel and you will be up to your ears in options.
I second the 2010 adaptation. It is pretty grounded, unlike many over-the-top adaptations of Romance of the Three Kingdom which are more well-known in the west.
And the performance of Cao Cao is great