Vivek has done an ok job of correcting this, but since you asked, here's my corrected version too :-)

My first trip to New York! (Part one)

One day I looked in map book an atlas, and then I came to think of realize I have never been outside Europe. Then I saw New York. It´s a long distance, so it´s could be a nice and a big adventure, hopefully. You hear about NY all the time on TV, and read about it in the newspaper. It´s a very popular destination! And as I have become more interested in English, I decided to go. I traveled with my mom. First, we went to a travel agency. They gave us information about different hotels and things to do there. My longest flight this far was when me and my girlfriend* were traveling to an Island in Greece. It was 25 years earlier...
When a the plane takes off, I always feel a bit nervous in my stomach.** On the plane there was were screens that allowed you to watch movies and listen to music. We got food at least two times, and the food was good. When we arrived at Newark airport and got off the plane, we were looking at the letters  saw the message: "Welcome to the United States of America", and that was a great feeling, and it felt really big! The local time was around 4-5 pm, I think. Our travel guide told us to drink a lot of water and go to bed early. This because of the Jet lag (the time difference).
A new feeling appeared realization dawned on me when we arrived to at our hotel. I could use my English at the reception desk, and so I did. It worked well. Our room was good. When we have had a look at it, we decided to get something to eat. It was raining so we picked a restaurant nearby the hotel. I ate a hamburger with french fries. Our breakfast favorite became "Pret a manger". This organic food chain had it all; oatmeal, orange juice, hot chocolate, croissants, bagels with cream cheese and more. It tasted great! We both liked it very much. After having our breakfast we walked around in the city. We looked at skyscrapers and tried to make us comfortable acclimate ourselves. In our trip there was a sightseeing tour included. It was a good start, even if we were a little late and our taxi arrived just in time... We traveled north from midtown to Harlem via Metropolitan Opera. On the way back from Harlem, we drove on the right side of Central Park. We saw two famous museums before coming to the exclusive shops at fifth avenue. Then we continued to downtown and battery park. Here we jumped off the bus to see the Statue of Liberty. But the distance was quite a long way from there. A few days later we took one of the yellow ferries to Staten Island to come get closer and have a closer look at it. (before the trip I bought a pretzel. It tasted very good!) End part1

*Technically it should be, "when my girlfriend and I were traveling...", however it is actually very common in American English to say it exactly how you said it. It's not "correct," but we still say it that way all the time.

**A common way to say this: "I get butterflies," (Meaning that there are butterflies inside your stomach)

My first trip to New York! (Part one)

One day I looked in the map book (better: I was going through the map book), and then I realized I had never been outside Europe. Then I saw New York. It was a long distance, so it would be a nice and a big adventure, hopefully. You hear about NY all the time on TV, and read about it in the newspaper. It´s a very popular destination! And as I had become more interested in English, I decided to go. I traveled with my mom. First, we went to a travel agency. They gave us information about different hotels and things to do there. My longest flight till then was when my girlfriend and I were traveling to an Island in Greece. It was 25 years ago...
When a plane takes off I always feel a bit nervous in my stomach. On the plane there were screens (better: the plane had screens...) that allowed you to watch movies and listen to music fg. We got food at least two times, and it was good. When we arrived at Newark airport and got off the plane, we were looking at the words (letters could be used if it's only one word): "Welcome to the United States of America", and that was a great feeling, and felt really big! The local time was around 4-5 pm, I think. Our travel guide told us to drink a lot of water and go to bed early. This because of the Jet lag(the time difference)
A new feeling appeared when we arrived to our hotel. I could use my English at the reception desk, and so I did. It worked well. Our room was good. After having a look at it, we decided to get something to eat. It was raining so we picked a restaurant near the hotel. I ate a hamburger with french fries. "Pret a manger" became our breakfast favorite. This organic food chain had it all: oatmeal, orange juice, hot chocolate, croissants, bagels with cream cheese and more. Everything tasted great! We both liked it very much. After having our breakfast we walked around in the city. We looked at skyscrapers and tried to make ourselves comfortable. In our trip there was a sightseeing tour included. It was a good start, even though we were a little late and our taxi arrived just in time... We traveled north from midtown, to Harlem, via Metropolitain Opera. On the way back (from Harlem), we drove on the right side of Central Park. We saw two famous museums, before coming to the exclusive shops at fifth avenue. Then we continued to downton and battery park. Here we jumped off the bus to see the statue of Liberty. But it was quite far/But it was a long distance from there. A few days later we took one of the yellow ferries to Staten Island to get closer and have a closer look at it. (before the trip I bought a pretzel. It tasted very good!) End part1

A tip: when you are narrating in one tense, keep writing in the same tense. For example, if you choose to narrate events in the past tense, then stick with past tense (unless you want to talk about something in the present, which is not a part of the narration; you can change the tense then).

All in all, a very good job!