Сегодня такой сильный ветер, что я не хочу выходить из дома.

This is a very common Russian pattern meaning so ... that.

• такой сильный ветер = such a strong wind
• что я не хочу выходить из дома = that I don’t want to go out of the house / leave home

So the whole structure is:

• такой + adjective + noun, что ...
• so / such a ... that ...

You can think of it as:

• Сегодня такой сильный ветер, что...
• Today the wind is so strong that...

Russian often uses такой where English may use either such or so depending on the sentence.

In Russian, the verb to be is normally omitted in the present tense.

So:

• Сегодня такой сильный ветер literally looks like Today such strong wind
• but it means Today there is such a strong wind or The wind is so strong today

This is normal Russian grammar. Compare:

• Он студент. = He is a student
• Она дома. = She is at home
• Сегодня холодно. = It is cold today

In past and future, forms of быть do appear:

• Вчера был сильный ветер. = Yesterday there was a strong wind
• Завтра будет сильный ветер. = Tomorrow there will be a strong wind

Because сильный describes ветер, and it must agree with it in gender, number, and case.

Here:

• ветер is masculine
• singular
• nominative case because it is the subject

So the adjective must also be:

• masculine
• singular
• nominative

That gives сильный.

Compare:

• сильный ветер = strong wind
• сильная буря = strong storm
• сильное течение = strong current
• сильные ветры = strong winds

For the same reason as сильный: такой agrees with ветер.

Since ветер is masculine singular nominative, the matching form is такой.

Compare:

• такой ветер = such a wind
• такая погода = such weather
• такое утро = such a morning
• такие дни = such days

So in такой сильный ветер, both такой and сильный match ветер.

Because it is the main noun being talked about: the wind is what is strong today.

In the first clause:

• Сегодня такой сильный ветер

the noun ветер functions as the subject-like center of the statement, so nominative is used.

You can think of it as:

• There is such a strong wind today or
• The wind is so strong today

Russian often expresses this without a separate verb in the present tense, but the noun still stays in the nominative.

Yes, Russian word order is flexible. Сегодня is at the beginning because it sets the scene right away: Today...

You could also hear:

• Такой сильный ветер сегодня, что я не хочу выходить из дома.
• Я не хочу выходить из дома: сегодня такой сильный ветер.

The original sentence sounds natural and neutral. Putting сегодня first emphasizes the time frame first.

So the beginning position is not required by grammar; it is mainly about focus and style.

Here что means that and introduces the result clause.

The sentence has two parts:

1. Сегодня такой сильный ветер
2. что я не хочу выходить из дома

Together they mean:

• Today the wind is so strong that I don’t want to leave the house

So this is not the question word what. It is the conjunction that.

This is a standard pattern in Russian:

• Так холодно, что я не могу спать.
• Он так устал, что сразу лёг.
• Был такой шум, что мы ничего не слышали.

This is about aspect.

• выходить = imperfective
• выйти = perfective

In this sentence, выходить sounds natural because the speaker means I don’t want to go out / leave home in general in these conditions, not just one single completed act.

So:

• не хочу выходить из дома = I don’t feel like going out / leaving home

If you said:

• не хочу выйти из дома

it would sound more like I don’t want to step out once or I don’t want to go out at this specific moment as a single action.

Both can be possible in some contexts, but выходить is the more natural choice for a general unwillingness to go out because of the weather.

Из requires the genitive case, so дом becomes дома.

• дом = house, home
• из дома = out of the house / from home

So:

• выходить из дома = to go out of the house, to leave home

This is a very common preposition-case combination:

• из школы = from school
• из города = from the city
• из комнаты = out of the room

So дома here is genitive singular of дом.

Not in this sentence.

This is a common source of confusion because дома can mean two different things:

1. дома as an adverb = at home

   • Я дома. = I am at home.

2. дома as the genitive singular of дом

   • из дома = out of the house / from home

In your sentence, because it comes after из, it is clearly the second one:

• из дома = from home, out of the house

So here дома is not the adverb at home.

Yes. Russian has no articles like a and the, so ветер can mean:

• wind
• the wind
• sometimes even a wind, depending on context

In this sentence, English usually translates it as:

• Today the wind is so strong that... or
• Today there is such a strong wind that...

Russian does not need an article to show that.

Because Russian uses хотеть very naturally for this idea.

• я не хочу выходить literally = I do not want to go out
• but in natural English, depending on context, that may be translated as:
  • I don’t want to go out
  • I don’t feel like going out
  • I don’t want to leave the house

So the Russian is straightforward:

• хочу + infinitive = want to do something
• не хочу + infinitive = don’t want to do something

Examples:

• Я хочу спать. = I want to sleep.
• Я не хочу работать. = I don’t want to work.
• Я не хочу выходить. = I don’t want to go out.

The main stresses are:

• сегóдня
• такóй
• си́льный
• ве́тер
• что
• я
• не хочý
• выходи́ть
• из дóма

A learner-friendly pronunciation guide:

• сегóдня = see-VOD-nya
• такóй = ta-KOY
• си́льный = SEEL-nyy
• ве́тер = VYE-ter
• хочý = kha-CHOO
• выходи́ть = vy-kha-DEET'
• дóма = DO-ma

Stress is important in Russian because it can move around and is not always predictable.

Yes, absolutely. That is a very natural translation.

Possible good English versions include:

• Today the wind is so strong that I don’t want to go out.
• There is such a strong wind today that I don’t want to leave the house.
• The wind is so strong today that I don’t want to go outside.

The exact English wording can change a little, but the Russian grammar stays the same.