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I've also noticed that other words can be before or after the subject:
What is the difference in nuance of placing such words before or after the subject, if any?
It's really a matter of emphasis. The greater emphasis is on the word that goes first.
今天我去..., means, Today, I am going out...
On the other hand, with 我今天去...,
the emphasis is on the 我 or I.
As an independent sentence, there is no difference between the two sentences. but if there is context, things could be different.
if you replace "今天我去" with "我今天去"，the feeling is different.
I guess this kind of grammar is not unique only in Chinese.
generally it's most likely that it's the same with English.
can we say in English:
Today I went to the cinema.
I today went to the cinema.
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