That year, I washad just been transferred to a new department to be in charge of, and there was a work meeting on Huangshan soon after.

(1) You can consider using some commas to break up this lengthy sentence and make it easier to read.
(2) The past perfect tense "had just been" might be more appropriate here, since your transfer is a past event that occurred before another past event (your visit to Huangshan).
(3) The phrase is "be in charge of (something)".

When we arrived at Huangshan and got out of the bus, many people followed us to asking us to book rooms in their hotels, which made us upset and wannoyed. We put a lot of effort into getting rid of these peoplem.

(1) I replaced "to ask" with "asking" to avoid repeating the "to" in such close proximity.
(2) I don't think "upset" is the best word to use here. I might suggest "annoyed".
(3) I think splitting this sentence into two would sound more natural.
(4) I would replace "these people" with "them", since you'd already used "people" not long before this (in "many people followed us").

Since we were not richdidn't have that much money back then, we didn't take the cableway to climb the mountain; we onlyjust hiked the mountainup it.

(1) "We were not rich back then" implies that you are rich now, which can potentially come across as boastful. I have rephrased it so that it's much more modest.
(2) I feel "just" might be more appropriate than "only" here. Explaining it might be a little challenging though.
(3) The phrase is "hike up (a mountain)."
(4) I replaced "the mountain" with "it" to avoid repetition of "the mountain".

On our way up, there was a narrow path.

This sentence feels incomplete. It would be good to specify where you are heading towards.

Below it there was a deep cliffabyss.

"Cliff" refers to the rock face, while "abyss" refers to the hole or chasm. A cliff is steep, while an abyss is deep. ("Abyss"就是“渊”的意思。)

Many people stopped there and shouted loudly to express their amazement but didn't movecontinue moving forward.

Your original sentence is fine, but here's an alternative.

I realized some potential risks and shouted ato them to leave that place quickly.

We saw the "fFlying sStone", Yuping Llou, the Guest - -Greeting Pine and some other very famous scenic spots.

(1) "Flying Stone" is the name of a landmark, so it might be more appropriate to capitalise it. I think some people also call it the "Feilai Stone" (which is just directly translated from Chinese).
(2) "Yupinglou" seems to be spelt as one word.

It was then that I realtruly understood anthe old Chinese saying, "iIt's easy to hike up a mountain but difficult to get down."

(1) "Truly" might be more appropriate here.
(2) "The" is more natural here, as you're referring to a specific saying.

It seemed like it was the first time I had visited a tourist attraction with my wife and (my) son.

(1) Alternatively: "It seemed to have been the first time". I think I personally prefer this expression.
(2) You may omit the second "my".