חשוב שהילדים יתנהגו יפה גם בבית וגם בבית הספר.

A fairly literal breakdown is:

• חשוב = important
• ש־ = that
• הילדים = the children
• יתנהגו = will behave / should behave
• יפה = nicely / well
• גם בבית וגם בבית הספר = both at home and at school

So the whole sentence is structurally:

Important that the children behave nicely both at home and at school.

In natural English, we usually add it is at the beginning: It is important that the children behave nicely both at home and at school.

Because this is an impersonal expression.

Hebrew often says חשוב ש... for it is important that.... There is no separate word for the English dummy subject it here, so Hebrew uses the default form חשוב, which is masculine singular.

So:

• חשוב ש... = it is important that...

The whole clause שהילדים יתנהגו יפה... is the thing that is important.

ש־ means that.

It introduces a subordinate clause:

• חשוב = important
• שהילדים יתנהגו יפה = that the children behave nicely

This ש־ is extremely common in Hebrew and is often attached directly to the next word in writing.

This is a very common Hebrew pattern.

After words like:

• חשוב = important
• צריך = necessary / need to
• רצוי = desirable
• עדיף = preferable

Hebrew often uses the future tense to express something that is wanted, expected, recommended, or necessary.

So יתנהגו here does not necessarily mean simple future time. It often has a meaning closer to:

• should behave
• are to behave
• behave

So חשוב שהילדים יתנהגו יפה is naturally understood as: It is important that the children behave nicely.

יתנהגו comes from the infinitive להתנהג, meaning to behave.

It is:

• 3rd person
• masculine plural
• future

So it means:

• they will behave
• or, in this kind of sentence, they should behave / behave

It matches הילדים, which is grammatically masculine plural.

Because יפה is functioning like an adverb here.

In English, we say:

• nice children
  but
• behave nicely

Hebrew often uses an adjective form adverbially. So:

• ילדים יפים = beautiful / nice children
• יתנהגו יפה = behave nicely / behave well

Since יפה is modifying the verb יתנהגו, it does not agree with הילדים in number or gender.

הילדים means the children.

The definite article ה־ is used when the children are understood as a specific group or a group already known from the context.

So:

• ילדים = children
• הילדים = the children

In this sentence, the speaker is not talking about children in a totally general, indefinite way, but about a known or contextually understood group.

גם ... וגם ... is a very common Hebrew pattern meaning:

• both ... and ...
• or also ... and also ...

So:

• גם בבית = both / also at home
• וגם בבית הספר = and also / and both at school

Together:

גם בבית וגם בבית הספר = both at home and at school

This repeated גם is perfectly normal Hebrew.

Because בית הספר is a construct phrase (called סמיכות in Hebrew).

Literally, it is:

• בית = house
• ספר = book

So the literal historical meaning is something like house of the book, but in normal modern Hebrew it simply means school.

In Hebrew construct phrases, the definiteness shows up on the second noun, not the first.

So:

• בית ספר = a school / school
• בית הספר = the school

That is why the ה־ appears on ספר, not on בית.

Here בבית means at home.

The preposition ב־ means in / at, and it attaches directly to the word after it.

Depending on context, בבית can mean:

• in the house
• at home

In this sentence, the natural meaning is clearly at home.

So:

• גם בבית = both at home
• וגם בבית הספר = and at school

It can mean either, depending on context.

Literally, it is in/at the school, but in many everyday sentences it is best translated as at school.

Here the sentence is about behavior in different settings, so בבית הספר is most naturally:

at school

not necessarily specifically inside the physical building.

Yes, Hebrew can sometimes use an infinitive after חשוב, but the structure changes a little.

For example:

• חשוב שהילדים יתנהגו יפה = It is important that the children behave nicely
• חשוב לילדים להתנהג יפה = It is important for the children to behave nicely

Both are possible, but ש + future is especially common when the sentence presents a full clause with an explicit subject, as it does here.