Part II.
I couldn't finish it because it was time for me to go with my girlfriend to the Physiotherapist, due to a sudden pain that was ails her on a knee, (just a little thing, It wasn't serious, the Physiotherapist said).
*...physiotherapist due to a sudden pain that was ailing her on a knee (just a little thing; the physiotherapist said that it wasn't serious).
(("Physiotherapist" doesn't need to be capitalized since it isn't part of a title or a name. After "just a little thing" I might try a semi-colon rather than a period, but it still seems like it could be wrong to me. You might want to do some research about paranthesis, because I just don't know much about proper grammar inside of parantheses! This was more of a guess, so take it with a grain of salt :-]))

After that, we went to a multicultural fair, that was occurring in the mainly square of my city, it is called "Feria de las culturas amigas", something like "Friend culture fair".
*...multicultural fair that was occurring in the main square of my city. It is called...
((These were two independent clauses ("we went..." and "it is..."), so use the proper methods to split these up :-]. Also, you do not need a comma after 'fair' and before 'that.'))
*...multicultural fair that was being held in the main square of my city.
((Another phrase option))

There were a lot of stands, each one represented a country.
*There were a lot of stands. Each one represented a country.
((Split up the independent clauses by using a period or one of the other methods.))
*There were a lot of stands with each one representing a country.
((I want to say that there should be a comma after "stands" and before "with" since these are two independent clauses, but it looks wrong to me, and I am not sure if the word "with" gives some exception. It is probably safer to use a comma though, since that is the typical rule.))

Notwithstanding portions were small (and a little expensive If we thought they were tiny), I felt my belly was about to exploit, and my girlfriend was as satisfied as I was.
*Though the portions were small (and a little expensive considering how tiny they were), I felt my belly was about to explode, and my girlfriend was as satisfied as I was.
((Awesome punctuation!))

We were happy although, because we could learn about a lot of things of many countries and peoples and how they live.
*We were happy though, because we could learn about a lot of things of many countries and people and how they live.
*We were happy though, because we could learn many things about a lot of different coutries and about how their peoples lived.

We are vegetarians, so there were many dishes we didn't try, but we were very happy with the different ways water is flavored in some countries, for instance in Syria they do a kind of roses and milk infusion, it’s more than delicious, it’s the pure heaven on earth.
*We are vegetarians, so there were many dishes we didn't try, but we were very happy with the different ways water is flavored in some countries. For instance, in Syria they do a kind of roses and milk infusion; it’s more than delicious--it’s the pure heaven on earth.
((Awesome! Just keep practicing with independent clauses (subject + verb). If you have too many subjects and verbs in one sentence, it can be difficult for your reader to follow along.))

Then I took my girlfriend to her house and made my way back home. I finished reading a novel called “Yellow Birds” by Kevin Powers. It’s amazing! It’s about a solider that went to fight to Iraq’s war, and how he lives with the things he did there and the things (and people) he lost there. I strongly recommend it.
((Wonderful! Very well done!))

At the end I studied some cards I made with English words in order to recall those words and started writing this note.
*Afterwards, I studied some flashcards in English in order to learn those words and started writing this note.

Awesome job, Ernesto. This was really cool to read, everything was very clear, and I loved the picture you added at the end! I can't wait to read more! :)

I am going to delete any sentences that are fine, so you can more easily find the corrections and any explanations I make. :-)

Part I.

I woke up at 10 AM, unfortunately I slept fitfully, and therefore I was a little tired when my alarm clock went off.

*I woke up at 10 AM. Unfortunately, I slept fitfully; and therefore, I was a little tired when my alarm clock went off.

((This sentence actually contains three independent clauses [three phrases that have a subject and a verb and can be their own sentences]. "I woke...", "I slept...", "I was..." You can split up these independent clauses into three sentences or use different kinds of punctuations and appropriate vocabulary. One way to properly punctuate independent clauses (with subject + verb) aside from simply using periods is to use a semicolon{;} followed by therefore, however, consequently, and several other words. Another way is to use a comma {,} followed by something jokingly referred to as FANBOYS {for, and, nor, but, or, yet, so}. Use these methods to combine independent clauses. {There are also dependent clauses, so you might want to look at those too if you get the chance :-)}))

*I woke up at 10 AM. Unfortunately, I slept fitfully and therefore I was a little tired when my alarm clock went off.

((In this sentence I am combining your last two independent clauses and eliminating the word "I" {your subject} in the last independent clause. This means I don't need a comma {,} before the word 'and.' You also can just say 'alarm' without saying 'clock,' and English-speakers will know what you mean.))

*I woke up at 10 AM. Unfortunately, I slept fitfully, so I was a little tired when my alarm clock went off.
((You can combine these last two independent clauses and keep them as independent clauses if you use a semi-colon {;} or a comma {,} with FANBOYS {for, and, nor, but, or, yet, so}.))

That's why first thing I did was went to the kitchen and do some cafe.

*That's why the first thing I did was go to the kitchen and make some coffee.

((Notice here that you only have one subject, so you don't need to use a comma and FANBOYS or a semi-colon or anything fancy. Your punctuation is awesome here!))

Cafe makes me live my day in a most vivid way.

*Coffee makes me live my day a most vivid way.

(("a most vivid way" sounds like older English like Jane Austen, but it is correct and sounds nice. If you want to make it more modern try the following.))

*Coffee makes me vividly come alive.

*Coffee makes me live vibrantly throughout the day.

*Coffee makes me live my day full of energy.

I was supposed to practice my English and teach some Spanish, but my language partner never turned up in Skype.

*...on Skype.

((Awesome punctuation and grammar! Two independent clauses united! Nice casual language as well.))

So I started working in a song I'm trying to cover, a Gary Numan's song.

*So, I started working on...