I'm starting to write a diary from todaynow on.

This sounds a bit more natural

Currently, I'm learning English, so I want to write correct English sentences.

My diary will be included about my daily studying of English, my family and activities.

include and about kind of refers to the same thing so do not usually come together

I really hope this output beingis helpfull for my learning.

Now, I'm relaxing weeken that it's the weekend, I'm more relaxed.

I have done a few erreands, already.

I dropped off and picked up my daughter tofor dance lesson, then picked her up later.

My daughter is doing maths text,her math assignment now.

She seems to enjoying to study learning new knowledge.

I'm starwriting to write a diary froma diary starting today.

"From today" sounds a bit wrong in this context, so I reordered the sentence so it makes more sense.

Currently, I'm learning English, so I want to write correctaccurate English sentences.

"Correct" usually refers to getting a question right, accurate makes more sense in this context. Also, because you are refering to mulitple sentences, you add an s and make it plural.

My diary will be included about my daily studying of English, my family and other activities.

"Be included" doesn't make sense, so make it a verb form. Also, and_ activities sounds off because there were descriptions of every other item, so I added an other.

I really hope this output being helpfull for my learning.

"Helpful" is spelled with one l. This is generally true for all other words like this (beautiful, plentiful, etc.).

Now, I'm having a relaxing weekend.

"am" is a helping verb, so you need a stronger verb before you get to the noun. Additionally, "am" means you are ---, so it doesn't make sense here.

I have doneran a few erreands, already.

Just a few mispellings here! Also, usually you say that you "run an errand" instead.

I droped off and picked up my daughter tofor her dance lesson.

To doesn't make sense in respect to pick up, so just say for.

My daughter is doing maths text, homework now.

Text may mean a passage or writing, but we usually don't use it in respect to homework.

She seems to enjoying to studying new knowledgethings.

Thanks for reading, and I wish to get some feedbacks!

Feedback is always singular, you never add an s.

First Diary. Entry

I'm startgoing to write a diary entry starting from today.

Currently, I'm learning English, so I want to be able to write correct English sentences.

My diary will be includedtalk about my daily studying of English, my family and my activities.

I really hope this output beingwill be helpfull for my learning.

Now, I'mFor now, I will be relaxing for the weekend.

I have already done a few erreands, alrady.

I dropped off and picked upthen fetched my daughter tofor her dance lessons.

And II had also cooked yakisoba (Japanese fried noodle dish) for lunch.

My daughter is doing maths texther math homework, now.

(This is my best guess at what you meant to say.)

She seems to be enjoying to study new knowledgeing and learning new things.

Thanks for reading, and I wish to get some feedbacks!

"Feedback" is a mass noun, so it doesn't take a plural ending.

First Diary Entry.

A first diary means the first book that you are writing in

I'm starting to write a diary from today.

You don't need a preposition with "today"

Currently, I'm learning English, so I want to write correct English sentence.

My diary will be included information about my daily studying of English, my family and activities.

Include can be in the present tense, I would also use the preposition about to describe the subjects you will write about

I really hope this output beingwill be helpfull for my learning.

will be shows that the action will happen in the future
helpful is spelled with only one l at the end of the word

Now, I'm having/enjoying a relaxing weekend.

we would use a verb in this sentence

I have done a few erreands, already.

a few spelling issues, errands is spelled with an "a", already is spelled with an "e",
you don't need a comma before already

I droped off and picked up my daughter to dance lesson.¶
I have dropped off my daughter for her dance lesson and picked her up

My daughter is doing maths text, now.¶
My daughter is doing her math homework now

you don't need a comma before "now"

She seems to enjoying to study new knowledge. learning new things

Thanks for reading, and I wishhope to get some feedbacks!

Today, I'm starting to write a diary from today.

Currently, I'm learning English, so I want to write correct English sentences.

My diary will be included about my daily studyingies of English, my family and activities.

I really hope this output beingdiary is helpfull for my learning.

Now, I'm relaxing on the weekend.

I have already done a few erreands, alrady.

I droped off and picked up my daughter tofrom her dance lesson.

And I cooked yakisoba (Japanese fried noodle dish) for lunch.

MNow, my daughter is doing maths text, nowwork.

?

She seems to enjoying to studying new knowledge.

Thanks for reading, and I wish to get some feedbacks!

Please revise my articlecorrect my journal if you are free.