Starting this afternoon, we began our holiday for the Chinese Dragon Boat Festival began.

The original sentence has a mixed tense ("Starting" + "began") which creates confusion. Changing "we began" to "began" places the action clearly in the past while keeping the time reference.

It will last untill next Monday, June 2nd.

"Until" is more formal and standard than "till" in written English, especially in formal contexts. Also, "till" is often considered more informal or colloquial.

Usually, I would go hiking with my friends during holidays, but this time I'’ll stay in Shanghai for the wholeall three days because it'rain is expected to rain starting tomorrow night.

"Would go" suggests habitual past, but since this is a current general habit, "I go" is better.
"The whole three days" sounds awkward; "all three days" is smoother.
"It's expected to rain" is correct but passive; "rain is expected" is more natural here.

There is a rowing team ion the river near my home.

In English, we say "on the river" rather than "in the river" when referring to boats or activities happening on the water surface.

I'’ll go to watch their training tomorrow morning.

"Go watch" is a natural and common expression; the preposition "to" is unnecessary here.

Speaking of videos, I noticed that some of the videos on my YouTube channel weare gainetting more and more views, which madekes me feel a little bit stressed.

Present tense ("are getting") fits better here because the process is ongoing.
"A little bit stressed" can be shortened to "a little stressed" for conciseness.

But I think it'’s a good kind of stress for me.

"A good kind of stress" is a more natural English phrase; simply "a good stress" sounds awkward.

I need to capture Hhigher-quality videos and improve my English to a higher level.

"Higher-quality" should not be capitalized mid-sentence.

SMOOTHER: Starting this afternoon, we began our holiday for the Chinese Dragon Boat Festival holiday.

Usually, I would go hiking with (my) friends during these holidays, but this time I'll stay in Shanghai for the whole three days because it's expected to rain starting tomorrow night.

NOTE: The word "my" can be deleted.

But I think it's a good stress for meALSO POSSIBLE: However, this is a positive stress.

I need to capture Hhigher-quality videos and improve my English to a higher level.

It will last tillhrough next Monday, June 2nd.

Your original is grammatically correct, but due to the ambiguity of "till/until" (to some native English speakers, it means the festival ends at the start of June 2nd, while to others, it ends at the end of that day), I suggest using "through" instead. This way, it's clear that the 2nd is also a holiday.

Usually, I would go hiking with my friends during holidays, but this time I'll stay in Shanghai {for the whole three-day period | for all three days} because it's expected to rain starting tomorrow night.

Alternative: Usually, I would go hiking with my friends during a holiday like this, but....

There is a rowingdragon boat team in the river near my home.

Technically, it's paddling, not rowing, although many native English speakers confuse the two. :-)

If their boats are decorated for the Dragon Boat Festival, I'll take some videos for you.

I would enjoy seeing them paddling even if the boats aren't decorated. :-)

I need to capture Hhigher-quality videos and improve my English to a higher level.

Starting tThis afternoon, we began our holiday for the Chinese Dragon Boat Festival.

"starting" sounds slightly unnatural in this sentence. Typically you would say "starting" when stating a defined period of time e.g. "Starting from XX until XX", "Starting this XX and ending this XX". In your case you could say "Starting from this afternoon until Monday June 2nd, we have our holiday for the Chinese Dragon Boat Festival."

Usually, I would go hiking with my friends during holidays, but this time I'll stay in Shanghai for the whole three days because it's expected to rain startingfrom tomorrow night.

Minor point... we would only say "the whole three days" for emphasis on the time period. "The three days" would be to just state the time period normally.

I need to capture Hhigher-quality videos and improve my English to a higher level.

No need to capitalise higher

I hope you all have a happy Dragon Boat Festival, wherever you are.