My first note on Italki note book (people will usually understand if you write it this way but "notebook" is usually how it is written).

It's only 6 am. I woke up very early today, (a comma before "and" is not necessary, usually only before "but") and I seat am sitting in a front of my labtop, and write writing this.
Nice to meet you strangers (this is not wrong, however it might make people feel a little weird if you call them "strangers" instead of "friends"). I'm a Korean (you can also say "I am a Korean person"). I live in Seoul, the capital of South Korea. It's one of the trendy trendiest cities ("cities" is used instead of "city" since it is being compared to many cities in the world) city of the world. I had worked at a fashion trading company for 2 years, (the comma usually goes before the word "but") but, 2 months ago I quit, and now I'm looking for a new job. For those breaking time During this break, I traveled to the East Coast and Cebu. It was very memorable experience. I love to travel. so if I have a chance to go abroad or travel domestically, I usually don't hesitate (to leave go). Just go. So I hope to get a new job which that is related to travel. Anyway, thank you for reading my note. It was This is my first time, so it is very clumsy and unorganized... but could you check my note? Thanks.

**Overall, your English is pretty good. Continue studying and your English will definitely improve very soon. You can let me know if you have any questions about anything specifically and I will be able to help you. Oh, I hope you are able to find a good job and can keep traveling! 

My first note on italki notebook (notebook is one word).

It's only 6 am. I woke up very early today, and I seat a sat in front of my labtop laptop, (no comma here) and write wrote this.
Nice to meet you strangers. I'm a Korean. I live in Seoul, the capital of South Korea. It's one of the trendy city cities (when you use "one of the," you need to use a plural noun to follow) of the world. I had worked at a fashion trading company for 2 two (spell out numbers between 1-9) years, (comma goes before "but," not after) but 2 two months ago, I quit, and now I'm looking for a new job. For those breaking time During this break, I traveled to (see comment below) East Coast and Cebu. It was a very memorable experience. I love to travel. So if I have a chance to go travel abroad or domestically, I don't hesitate to leave. I just go. So I hope to get a new job which that is related to travel. Anyway, thank you for reading my note. It was my first time, so it is very clumsy and unorganized... (ellipses should have three periods) but could you check my note? Thanks.

I didn't want to change too much at once, but in the yellow highlighted portion, instead of saying "but two months ago, I quit," it would be less awkward to say "I quit two months ago."

In the green highlighted portion, the "the" is in question marks because it depends on what "East Coast" you're talking about. If you're talking about the east coast of Korea, then it should read "traveled to the east coast and Cebu." If East Coast is a specific name for a town/region/etc., then you would say "traveled to East Coast and Cebu."

Wow! I can't believe that this is your first time. Although there were some corrections, your overall sentence structure and grammar is very good. Some things to watch out for are: Don't forget words like "my" or "the" before nouns. Make sure that your tenses match up, so if you're talking about the past tense, make sure everything is in the past. I wrote out some of my longer comments below so it wouldn't break up the flow of the piece too much, but let me know if you have any questions!

My first note on italki note book.

It's only 6 am. I woke up very early today, and I sat in seat a front of my labtop, and wrote write this.
Nice to meet you strangers. I'm a Korean. I live in Seoul, the capital of South Korea. It's one of the trendy city of the world. I had worked at fashion trading company for 2 years but 2 months ago, I quit and now I'm looking for a new job. For this break those breaking time, I traveled to East Coast and Cebu. It was very memorable experience. I love to travel. So if I have a chance to go abroad or domestic, I won't don't hesitate to leave. Just go. So I hope to get a new job which related to travel. Anyway thank you for reading my note. It was my/the first time, so very clumsy and unorganized.. but could you check my note? Thanks.

i wish to find your dream job soon