What Education iIs

Proper capitalization in your title.

I saw a sentence, "Education is not the learning of facts, but the training of the mind to think," at the front desk of the educational institution at which my daughter is studying English, while I was waiting to pick my daughther up.

Add commas around the quotation for clarity.
"training of the mind" Article needed.
"pick her up" Avoid repetition.

After I picked up my daughter, I asked her if she knowsew the meaning of the sentence.

"knew" Past tense consistency

She read the sentence for a while and then told me that what was important was not the knowledge itself but how to think.

"what was important was" Correct word order.

I totally agreed with her.

"agreed" Past tense consistency with the narrative.

After arriving at home, she began to check and think about the mistakes she made in her exams.

"After arriving home" Cleaner phrasing.
"think about the mistakes" Correct collocation.

I thought it was funny but meaningful, lol.

Or you can say, "I thought it was funny but meaningful," for formal writing.
If you lean more on the casual tone of your text, then you can keep your original sentence.

I saw athe sentence "Education is not the learning of facts, but the training of the mind to think" at the front desk of the educational institution at which my daughter is studying English, while I was waiting to pick my daughther up.

"while" and "where" are both good here, "where" just focuses on the location more.
Personally I prefer "while" here because it emphasizes that you happened to see the sentence while doing something unrelated.

What Education iIs

I saw a sentence "Education is not the learning of facts, but the training of mind to think" at the front desk of the educational institution at which my daughter is studying English, while I was waiting to pick my daughther up.

Because you already have "my daughter" in the sentence, the reader will easily understand who you are referring to. This substitution also avoids having "my daughter" three times in two sentences.

After I picked up my daughter, I asked her if she knowsew the meaning of the sentence.

This verb is part of what happened in the past (picked up, asked, knew).

She read the sentence for a while and t. Then shen told me that it means what is important wais not the knowledge itself but how to think.

Here the present tense is used to report what she said because she is making a generalization (a statement about something that is generally true). She is not talking about a specific event in the past.

I separated her actions into two sentences to make it easier for the reader to understand quickly.

I deleted "the" before "knowledge" because the sentence is not referring to a specific body of knowledge; it is referring to all knowledge.

After arriving at home, she began to check and think about the mistakes she made ion her exams.

"Arriving home" is correct, but the wording is what we usually use for busses, trains, planes that follow a schedule. The word "arrive" is somewhat formal in English, and a native speaker would probably say, "After we got home." This phrase is widely used in all time frames: when we get home, if she gets home, when did you get home? will they get home? etc.

We think about something, or we think of something. A preposition is needed after "think"

What Education iIs

I changed to all capital letters as it is a title.

I saw aThe other day, I saw an interesting sentence. "Education is not the learning of facts, but the training of mind to think." It was at the front desk of the educational institution at which my daughter is studying English, whilere I was waiting to pick my daughter up.

This sentence is a bit too long so I broke it up a bit as well. But you can tweak it to your liking, as long as it's not too long.

But just "I saw a sentence" at the start is a bit awkward, so I added the time it happened and an adjective, to make it sound more natural. You can change the time and adjective to your liking.

After arriving at home, she began to check and think about the mistakes she made in her exams.