“Hamburg steaks!” my son answered as usual, but he also said something different after that,” l want to cook it.” ¶
Then he looked up for a recipe suitable for him, went out for shopping and got what we didn’t have in the fridge.

Not just doing what is “ready for you” to get a quick feeling of achievement, he really went through most preparations on his own, like measuring all the ingredients and understoodanding the process to thebeginning to end.

"understanding" to match the tense of "measuring" is more natural here.

“Hamburg steaks!” my son answered as usual, but he said something different after that,” l “I want to cook it.” ¶
Then he looked up for a recipe suitable for him, went out for shopping, and got what we didn’t have in the fridge.

The quotation marks around "I want to cook it" should start with a comma outside and open the quote correctly.
"l" should be capital "I".
The comma after "that" should be outside the quotation marks, not inside.
「I want to cook it」の引用符の使い方が不適切です。
「l」は「I」の間違いです。
カンマは引用符の外に置くべきです。
"looked up for" is incorrect; the correct phrase is "looked up."
"went out for shopping" is unnatural; "went out shopping" is the natural collocation.
Comma before "and" in a list improves clarity.
「looked up for」は間違いで、正しくは「looked up」です。
「went out for shopping」は不自然で、「went out shopping」が自然です。
リストでの「and」の前にカンマを入れると読みやすくなります。

Not just doing what is “ready for you” to get a quick feeling of achievement, he really went through most of the preparations on his own, like measuring all the ingredients and understoodanding the process to the end.

"most preparations" should be "most of the preparations" for grammatical completeness.
Parallel structure requires "measuring" and "understanding" to be consistent (both gerunds).
「most preparations」より「most of the preparations」が文法的に正しいです。
「measuring」と「understood」は形が違うため、両方を動名詞に揃えるべきです。

He struggled a little when he had to mince the onions bywith a hocho (a Japanese knife), but managed it.

When indicating an instrument, "with" is the correct preposition, not "by."
Add a comma before "but" to separate two independent clauses.
道具を示すときは「by」ではなく「with」を使います。
「but」の前にカンマを入れて文を区切ります。

Though I oversaw and helped him when necessary, the steak he cooked on his own was just perfect.

"Just perfect" is somewhat informal and redundant; "perfect" alone is clearer and more formal.
「just perfect」は口語的で冗長な表現なので、「perfect」だけのほうが明確で丁寧です。

He looked as if it was nothing, but I was very impressed inside.

"Inside" is unnecessary and sounds awkward in this context. "Very impressed" alone sufficiently expresses the feeling.
「inside」はこの文脈では不要で、不自然に聞こえます。「very impressed」だけで十分気持ちを表現できます。

“Hamburg steaks!” my son answered as usual, but he said something different after that,” l want to cook it.” ¶
Then he looked up for a recipe that was suitable for him, went out for shopping, and got what we didn’t have in the fridge.

Not justInstead of doing what is “ready for you” to get a quick feeling of achievement, he really went through most of the preparations on his own, like measuring all the ingredients and understood the process tountil the end.

He struggled a little when he had to mince the onions bywith a hocho (a Japanese knife) but he managed it.

He looked as if it was nothing, but I was very impressed inside.

Alternately : ". . . but deep down I was very impressed."

“Hamburger steaks!” my son answered as usual, but he said something different after that,” l want to cook it.” ¶
Then he looked up /for a recipe suitable for him, and went out for shopping andto goet what we didn’t have in the fridge.

"looked up" or "looked for"

NotInstead of just doing what is “ready for you” to get a quick feeling of achievement, he really went through most preparations on his own, like measuring all the ingredients and understoodanding the process to the end.

I think I'd say "Instead of just using what is pre-prepared for you", but I think what you've written is correct.

or "understanding the process completely".

He struggled a little when he had to mince the onions bywith a/using a hocho (a Japanese knife) but he managed it.