I'll start my vacation today.

Tomorrow is Ancestor's Day for Chinese people.

As my great-grandma (told me / once told me / used to tell me), the most important thing is to keep the ancestors in mind.

Just "told me" is fine, but "once told me" or "used to tell me" are a little more natural sounding here, depending on whether she told you it once or multiple times.

I will not return to my hometown to visit graves, eEven though I miss my great-grandma and my grandparents during this time every year, I will not return to my hometown to visit their graves.

Sounds better this way round. 'their graves', I assume you are talking about the graves of your grandparents and great-grandma.

Now, I'm sitting on a couch in a hotel nestled between mountains in Lishui, Zhejiang Province.

I've takenfilmed some videos and will show them to you when I get back to Shanghai.

'filmed' or 'recorded' sounds better with videos. 'taken' is more for photographs.

I will not return to my hometown to visit our graves, even though I miss my great-grandma and my grandparents during this time every year.

As my great-grandma told me, the most important thing is to keep theour ancestors in mind.

Now, I'm sitting on a couch in a hotel nestled between the mountains in Lishui, Zhejiang Province.

I will not return to my hometown to visit gravepay my respects, even though I miss my great-grandma and my grandparents during this time every year.

"Visit graves" is definitely understandable, but just not commonly used in English. Paying respect is one way to express it. Another potential translation is:
I will not return to my hometown to visit my family's graves. "Visit graves" by itself is a bit weird in English, because there is no strong culture of visiting ancestor's graves, so you need to explain whose graves you are seeing to make it sound a bit smoother.

Finally, in situations where you use "my" twice in a list, you can skip the second my.

I'm starting my vacation today.

Your text is correct, but sounds a bit off. You could also say "My vacation starts today."

I will not return to my hometown to visit graves, even though I miss my great-grandma and my grandparents duringevery year at this time every year.

Again, your proposal was not incorrect, but it sounds more natural this way.

As my great-grandma told me, that the most important thing is to keep theour ancestors in mind.

Now, I'm sitting on a couch in a hotel nestled between the mountains in Lishui, Zhejiang Province.