I'm tired from work.
It's fine outside the window.
I want to go on a trip somewhere.
The weather is fine outside.
→Your sentence is okay, this sentence is better.
I came home.
Today, I had a meal in the Spanish restaurant.
Paella was delicious!
Tonight, I had dinner at a Spanish restaurant.
The paella was delicious!
The air conditioner was broken several days ago.
I have to buy an air conditioner.
Because It's too cold in my room.
It seems to become colder next week.
Therefore I had to buy an air conditioner as it was too cold in my room.
→Changed into past tense.
It seems that it will become colder next week.
→Changed into future tense.
I must get ready to go out.
It's very cold outside.
I have to wear a lot.
It's very cold outside, so I have to wear a lot.
→Both sentences are correct, but this sentence denotes the reason as to why you "have to wear a lot," which is why I decided to link the two together in this correction.
Finally I came home.
It's 1:00 am.
There will be also an errand tomorrow, so I must get up early.
Finally I came back home.
→usually we use the whole phrase "come back home" to indicate the action.
It wasn't 1:00 am.
→past tense should be used - you can refer more contracted form usage on this website : http://www.englishcoursemalta.com/learn/list-of-contracted-forms-in-english/
I still have some work to do tomorrow, so I must get up early.
→sound more natural. "errand" is not usually used - "work" will be more easy to present in the oral/diary entry :)
One week began.
It is raining.
I'm tired and feel depressed.
I want to take off from work, but I have mountain of work as usual.
The first week of work has begun.
→"One week began" Does not make sense. What are you trying to say??
I'm tired and I feel depressed.
I want to take off from work, but I have a mountain of work to do, as usual.
I'm tired and feel depressed.
→You don't really need feel here, though it's by no means incorrect. "I'm tired and depressed" sounds more natural though.
I want to take off from work, but I have a mountain of work as usual.
→You need to add the article "a" before the noun. While this is not necessary in Russian, it is in English :)
My work is piling up.
I'd like to go home early and sleep.
But I have to work, so I don't return home.
I want to eat a cream puff.
A cream puff makes me happy.
→Although individually the sentences you wrote are not wrong, there are still ways to improve them as the they are all quite short. For example, you can combine them and say "I want to eat a cream puff as it makes me happy." and "Although I'd like to go home early and sleep, I can't return home as my work's piling up."
I got a rare item by having defeated a lot of blue dragons.
I defeated them all the time for these 4 days, so I'm tired very much.
I need a rest.
I got a rare item by defeating a lot of blue dragons.
I fought and defeated them all the time for these 4 days, so I'm very tired.
I need to rest.
I feel bad today, so I decided to be absent from a company.
Although My work piles up, myself is important for me than works.
I'll go to the hospital soon.
I feel bad today, so I decided to be absent from the company.
Although my work piles up, my health is more important for me than my works.
→this one was a little tricky. there's some awkward phrasing, and while using "myself" (which i corrected to "my health") is technically right, it just sounds weird.
Today, I got a CT scan of the brain in the hospital.
Although there was not the abnormality at all, It's certain I'm tired.
I'll go to a bedrock bath now.
Although there wasn't any abnormality at all, it made me really tired.
→Your idea doesn't really flow to the next point... Just a suggestion on changing it slightly :)
I was absent from the company to have a medical examination at the hospital today.
In the afternoon I went to renewal my driver's license.
Since I am in poor physical condition, I intend to sleep early.
I was absent from work today because I had a medical examination at the hospital.
→Suggest use 'work' instead 'from the company.
Use coordinating conjunction 'because' and the past tense 'I had' here
In the afternoon, I went to renew my driving license.
→Include comma after 'afternoon' .
suggest watch video on word order: https://www.youtube.com/watch?v=VPyo8-Pr55Q
'renewal' is a noun. Use verb 'renew' here.
'My driver's license'' does not sound right. Suggest use 'my driving licence'
If you use 'my driver' you actually mean 'a chauffeur'
Since I am in a poor physical condition, I intend to go to bed early.
→You need to use indefinite article 'a' before a singular noun 'condition'
Read this website: http://esl.about.com/cs/beginner/f/f_athe.htm ...... and if you have time try this
Suggest use 'go to bed' instead of 'to sleep' here
I traveled in Croatia and Slovenia last summer.
Although I actually wanted to go to southern Italy, I didn't match the touring schedule.
Those countries were also very fun.
I want to travel in southern Italy this summer.
I traveled to Croatia and Slovenia last summer.
Although I actually wanted to go to southern Italy, I couldn't go because my scheduled dates clashed.
→I'm not sure what you are trying to say, so I wrote something that may be what you wanted to convey.
But those countries were also very fun.
I want to travel to southern Italy this summer.
Tomorrow is Saturday and the company is closed.
I'll see "GIRLS und PANZER" in movie theater.
It's my 4th time seeing it.
Although this is the first time to see same movies four times, I don't get tired at all.
I'll see "GIRLS und PANZER" in the movie theater.
Although this will be the first time to see the same movie four times, I haven't gotten tired of it at all.