26.07.2012

Yesterday was my friend's birthday who came to Vanuatu with me.

Yesterday was the birthday of a friend who came to Vanautu with me.

However, he is staying in Japan because he needed to purchase his new computer instead of using broken old one.

instead of what?
instead of using?
instead of buying?
it's missing a verb.

I check his entries on Facebook everyday and he seems to enjoy his life. He especially enjoys seeing his friends in Japan.

As a change the subject, my English training session last night was quite good.

I felt the mental stimulation tire me like when I work out in the gym and I am physically tired.

The kinds of training that gives me some burden are often difficult for me to continue, but they are essential for us to master something.

After I finished it, I thought that I would like to do it every day for at least three months.

After I finish this, I think that I would like continue to do this everyday for at least the next three months.

I hope that I my listening ability will advance in three months.

However, he is staying in Japan because he needed to purchase his new computer instead of using his broken and old one.

Best to use 'and' to separate the adjectives.

I check his entries on Facebook every day, and he seems to enjoy his life, especially enjoy seeing his friends in Japan.

Masa is right on the every day, since "everyday" is an adjective and "every day" means 'each day'; it is more fitting to use "every day".

I tried to repeat a lot of sentences from my textbook every time the CD read one of them.

Do you mean textbook right? Text has a lot of meaning, and most westerner would think you were talking about テキスト・メッセージ. :P

I felt some burden in My brain felt tired like my muscles when I work out in the gym.

The burdening training that give me some burden are makes it difficult for us to continue, but they are it is essential for us to mastering something either.

Note that I'm using present tense here. Although you shouldn't switch tense in your writing, since this sentence is talking about training in general and have no-time relations; it is ok here to switch to present tense.

After I finished it, I thought that I would like to do it this training every day for at least three months.

I hope that I will advance my listening ability will improve in three months later.

You should use 'improve' in this case rather than 'advance', since it means 'to get better at something'. It's more specific and more fitting in this sentence.