Foreign Tongue

When he was a first grader, his text book was very simple-- large characters and only or mostly hiraganas.

But he's a second grader now and his text book of Japanese language text book has become much more advanced.

And one day, I realized that I should do this myself--for my English practice. (no "s")

I silent-read English books, listen to English news and so on, and write emails at work.

But (in terms of)/ (as far as)/ (as for) concerning practices of pronouncing English words, I've hardly done it so far.

These are options that sound a little more natural and relaxed, more in keeping with the tone of your essay.

So I decided to start with the very elementary one. something easy.

A long time ago, I bought a book titled "Frog and Toad Are Friends" (see photo 2).

I wanted my teacher or "mother" or "father" to do that.

To do what? This sentence is unclear. Maybe you mean: "But I needed a "mother" or "father" to read it to me."

Because I can't have one already, Since that wasn't possible, I bought a CD recording of the book. in which reading voices are recorded.

Or: Since that wasn't possible, I bought the book on CD.

The reading voice on the CD is very clear, relatively slow, and full of intonation.

While I was "shadowing" after the reading voice one day, I realized I was using my mouth muscles and tongue differently from my speaking in Japanese. (or: differently than when I speak Japanese)

My younger son has homework of reading aloud every day. [Alternatives: "Part of my younger son's homework each/every day is to read texts/passages aloud." "My younger son has (some) reading aloud for homework every day."]

When he reads, either my wife or I check his reading and give a score—◎for good, ○ for OK, and △ for needing more practices--and a signature. [Alternative: "-- and sign our names."]

"Give a signature" is a bit odd-sounding. "Sign our names" is unfortunately a bit awkward in this case, too, since presumably only one of you signs his or her name.
"Practice" is usually used as a non-count noun.

Sometimes, if the new story is difficult for him, his reading on the first day may be awkward.

I would advise choosing one or the other of "sometimes" and "may be". The combination is not wrong, but it does sound a bit redundant. "May be" can be replaced by "is".

And one day, I realized that I should do this myself--for my English practices.

I don't see anything wrong with the pronoun "my" here.

I read English books silently, listen to English news and so on, and write emails at work.

Many people do use "email" as a count noun. Personally, I prefer to think of it as a non-count noun, corresponding to the non-count noun "mail", or to write "e-mail messages". I am in the minority concerning this, however-- the use of "email" as a count noun meaning either "e-mail message" or "e-mail address" as required seems to be standard usage.

But I haven't done much (so far) to practice pronouncing English words.

This is an alternative; your sentence sounds a bit more like a Japanese sentence than an English sentence. (English sentences rarely have topics.)

So I decided to start with a very elementary book/text. [Alternative: So I decided to start with something basic.]

I wanted my teacher or "mother" or "father" to read it aloud to me.

Because I didn't have one already, I bought a CD recording of the book.

While I was "shadowing" after the reading voice one day, I realized I was using the muscles of my mouth and tongue differently from when I speak in Japanese.

"Tongue" has the meaning of "language" among its many meanings. [Alternative: 'The word "tongue" has...' 'One of the many meanings of the word "tongue" is "language".']

You are talking about one specific meaning of the word "tongue", so the definite article "the" is needed.