No need for me to enter into the edit/correction forray.  I won't be able to do much better than DesertJedi & Dragon8.  Only one comment: DJ had asked where Puno is.  You should tell your readers earlier where the place is.  You are not writing a mystery piece to keep your readers in suspense. Of course for a those who know the place, it is OK but you need to think of those who don't. :)  You are a 1st class world traveller.  From India and now the Andes!

The Tastiest Air in the World (the first half)

When I arrived at Puno, I was excited. There was the town where I could feel their long history, and the people who have national characteristics. It was the very world which reminded me the animated cartoon series “3000 Leagues in Search of Mother”. (Marco was traveling in Argentina though) It was easy for me to grow fond of the town.
I couldn’t help being frolicsome, I then tried to be calm by the guide words. He had said that “Don’t run! You must not hurry. Be sure to be slow, just slowly” Puno is located at 3885m above sea level, and center of the Andes Mountains. The air is thin at this high altitude. He said it to me several times when I had left for this town. “Slowly, just slowly”
His suggestion was always right. He called my meticulous attention to any trouble in Lima (the capital of Peru), since it is not quite safe town. He might anticipate that I would be frolicsome, and he worried that I would get mountain sickness.
At night, I couldn’t get sleep at all. Although I had confidence about my physical strength, I was dead tired. My body felt heavy. I had a throbbing pain in the head. I also got out of breath. Is it comes from the effect of thin oxygen? I went to the downstairs with expectation of thick oxygen. It was even regretful that I stayed on the third floor. I even would be happy if they gave me a bed in the underground warehouse. Their staff offered me a cup of coca tea. They said that it’s helps a lot for mountains sickness. I wasn’t sure which did it work the ingredient of coca or the placebo effects. I felt better a little.

To be conclueded

The Tastiest Air in the World (the first half)

When I arrived at Puno, I was excited. There This was the a town where I could feel their its long history, and where the people who have had national characteristics(？).

(？) I did not understand what you meant by "national characteristics"

It was the very world which reminded me of the animated cartoon series “3000 Leagues in Search of Mother”. (The only difference was that Marco was traveling in Argentina. though). It was easy for me to grow fond of the town.
I couldn’t help being frolicsome feeling excited (contain myself) so I then tried to be calm myself by remembering the guide's words. He had said that , “Don’t run! You must not hurry. Be sure to be slow down, just slowly .”

"To contain oneself" is an English expression that means to calm down when you are feeling very excited and want to jump/run/dance

I couldn't contain myself when I shook hands with my favorite actress!

Puno is located at 3885m above sea level, and is located at the center of the Andes Mountains. The air is thin at this high altitude. He said it to reminded me several times when I had left for this town. “Walk(？)Slowly, just walk(？) slowly” 

His suggestion advice was always right.

to suggest and to advise are similar verbs.Advise is a stronger kind of suggestion

"I suggest you try the banana flavored ice cream.  It's delicious!"

"I advise you to sleep early.  Tomorrow we have an important test."

 An advisor is a person who gives advice.

He called my meticulous attention to any notified me of trouble in Lima (the capital of Peru) since it is was not quite and is an unsafe town. He might anticipated that I would be frolicsome too lively(energetic), and he worried that I would get mountain sickness.
At night, I couldn’t get sleep at all. Although I had confidence about my physical strength, I was dead tired. My body felt heavy. I had a throbbing pain in the head. I also got out of breath. Is Did it comes from the effect of thin shortage of oxygen? I went to the downstairs with the expectation of thick more oxygen. It was even regretful I regretted that I stayed on the third floor. I even would be have been happy if they even gave me a bed in the underground warehouse. Their staff offered me a cup of coca tea. They said that it’s helps a lot for really helped cure/prevent mountains sickness. I wasn’t sure which did it work the ingredient of coca or the placebo effects. I don't know if it was the coca or a placebo effect, but I felt better a little.

To be conclueded concluded

 # Once again, I enjoyed reading your writing.  Keep up the good work!  I would like to go mountain climbing one day.

# ゆみーさんの文書も楽しく読んでもらいました。いつか、登山しようとしたいですよ。

The Tastiest Air in the World (the first half)

When I arrived at Puno, I was excited. There This was the a town where I could feel sense their long history and the cultural identity of the people who have national characteristics. It was seemed the very world which reminded me of the animated cartoon series “3000 Leagues in Search of Mother” (Marco was traveling in Argentina though). It was easy for me to grow fond of like the this town.
I couldn’t help being frolicsome feeling giddy! I then tried to be calm myself by the recalling my guide's words. He had said that “Don’t run! You must not hurry. Be sure to be go slow just slowly”. Puno is located at 3885m above sea level and at the center of the Andes Mountains. The air is thin at this high altitude. He said it to me several times when I had left was leaving for this town. “Slowly, just slowly Slow, go slow”.
His suggestion advice was always right. He meticulously called my meticulous attention to any trouble in Lima (the capital of Peru), since it is not quite a very safe town city. He might anticipated that I would be frolicsome giddy, and he worried that I would get mountain altitude sickness.
At night, I couldn’t get sleep at all. Although I had confidence about my physical strength, I was dead tired. My body felt heavy. I had a throbbing pain in the my head. I also got out of breath. Is it comes Is this from the effect of thin oxygen? I went to the downstairs with the expectation of thick oxygen breathing easier. It was even regretful that I had to stayed stay on the third floor. I even would be would have been happy if they gave had given me a bed in the underground warehouse. Their staff offered me a cup of coca tea. They said that it’s helps it helps a lot for with mountains altitude sickness. I wasn’t sure which helped more: did it work the ingredient of the coca or the a placebo effects effect. I felt a little better a little.

To be conclueded concluded...

Yumi, you never mentioned if Puno was in Peru. I assume it is. Many of my corrections are simply improvements to make your text sound more like a native English speaker. They are not necessarily mistakes.

Good work!!!!