On outside it was cold and raining.

ThaIt's autumn.

Autumn sadness like in the song Lana Del Rey.

A good emotive remark

Today I had bad mooda little bit of a bad mood.

Today my English practice was't a loti didn’t practice my english much.

I just want justo relax today and may be tomorrow aswell.

For more practice my Englishenglish practice. I want to tell you about my vacation in South Korea.

I arrived toin Busan.

Busan is the second biggest city in South Korea after Seoul.

Oh I forgot tell you about a beach and sea.

On outside was cold and raining.

This form also works

That's autumn.

Nice one

Autumn sadness, like in the song by Lana Del Rey.

Today I had bad mood little bitwas in a little bit of a bad mood.

Today myI didn't do a lot of English practice was't a lot.

This sounds more natural

I just did my homework from the textbook and wrote this journal.

I just want justo relax today and may be tomorrow.

Maybe is one word. I just want = the only thing

For moreIn order to practice my English, I want to tell you about my vacation in South Korea.

In order to means "for the purpose of". The original sentence sounded very strange

Busan is the second biggest city in South Korea after Seoul.

I visited the most popeular places in the city like the fish market, the shop, the square and other beautiful places.

When talking about specific locations, generally you will need to prefix 'the'

Oh, I forgot to tell you about athe beach and the sea.

The - again here

It'sThey were beautiful too.

You are talking about two objects - beach and sea, so they are beautiful, not it (one thing)

But the weather for was hard.

I'm not sure what you want this sentence to mean. Maybe:

"But the weather was bad" or "the weather was unpleasant"?

Veary hot and very wet.

May be it rained every day beacause I was in Korea induring the rainy season.

On outside it was cold and raining.

Today I hadwas in a bad mood little bit.

You can be 'in' a bad mood

Today my English, and I didn't practice was'tEnglish a lot.

It's better if you combine these two sentences into one sentence with two separate clauses. Writing 'Today; at the start of two consecutive sentences is not wrong, but still a bit untidy.

I just did my homework from a/the textbook and wrote this journal.

I want to just relax today and may be tomorrow.

For moreTo further practice my English, I want to tell you about my vacation in South Korea.

This would sound better

I arrived ato Busan.

Busan is the second biggest city in South Korea after Seoul.

I visited the most popeular places in the city, like the fish market, shops, square and other beautiful places.

Oh I forgot teo tell you about athe beach and sea.

It's better if you write 'the beach' instead of 'a beach'. 'A beach' implies that you are not sure which beach you are talking about, it could be any beach. But 'the beach' implies that you are referring to that specific beach that you visited.

But the weather for was hardwas unpleasant.

Maybe unpleasant would be more appropriate in this context than hard

Veary hot and very wet.

You don't need to write 'very' twice.

May be every day beacause I was in Korea induring the rain season.