רופא המשפחה שלי אמר שהחיסון הזה חשוב במיוחד בחורף.

This is a very common Hebrew structure.

• רופא = doctor
• המשפחה = the family
• שלי = my

So literally, it is something like doctor of the family, mine. In natural English, that becomes my family doctor.

Hebrew often expresses this kind of relationship with a construct chain:
רופא המשפחה = the family doctor

Then שלי is added after the whole phrase:
רופא המשפחה שלי = my family doctor

So unlike English, Hebrew does not usually put my before the noun here.

Because רופא המשפחה is a construct phrase.

In this kind of phrase, the first noun usually does not take ה־, even when the whole phrase is definite.

• רופא משפחה = a family doctor / family doctor in general
• רופא המשפחה = the family doctor
• רופא המשפחה שלי = my family doctor

So רופא stays without ה־, and the definiteness comes from המשפחה.

ש־ is a very common Hebrew word meaning that.

So:

• אמר = said
• שהחיסון הזה חשוב... = that this vaccine is important...

In full form, Hebrew can also use אשר, but in everyday Hebrew ש־ is much more common.

So:

• אמר שהחיסון הזה חשוב = said that this vaccine is important

In Hebrew, demonstratives like this and that usually come after the noun.

So:

• החיסון הזה = this vaccine
• הספר הזה = this book
• הילדה הזאת = this girl

Also, the noun is usually definite, so it takes ה־.

That is why Hebrew says:

• החיסון הזה

not:

• זה חיסון for this sentence

זה חיסון would mean something more like this is a vaccine.

They agree with חיסון, which is masculine singular.

• חיסון = masculine singular noun
• הזה = masculine singular form of this
• חשוב = masculine singular form of important

So:

• החיסון הזה חשוב = this vaccine is important

If the noun were feminine, the forms would change:

• התרופה הזאת חשובה = this medicine is important

This kind of agreement is very important in Hebrew.

במיוחד means especially.

In this sentence:

• חשוב במיוחד בחורף = especially important in winter

Hebrew word order here is very natural and close to English.
You can think of it as:

• important
• especially
• in winter

Hebrew is somewhat flexible with adverbs, but חשוב במיוחד בחורף sounds very normal.

Because it combines:

• ב־ = in
• החורף = the winter

Together, they become:

• בחורף = in the winter / in winter

This kind of combination is extremely common in Hebrew:

• בבית = in the house
• בכיתה = in the classroom
• בחורף = in the winter

So Hebrew often writes the preposition and the noun together as one word.

It can correspond to either one in English, depending on context.

Hebrew often uses the definite article with seasons more naturally than English does. So even though בחורף literally includes the, the best English translation may simply be:

• in winter

or

• in the winter

In your sentence, especially in winter is the most natural English translation.

Yes, very much so.

The sentence is:

• רופא המשפחה שלי = my family doctor
• אמר = said
• שהחיסון הזה חשוב במיוחד בחורף = that this vaccine is especially important in winter

So the overall structure is:

[subject] + [verb] + [that-clause]

That is very similar to English:

My family doctor said that this vaccine is especially important in winter.

So even though some smaller pieces work differently, the big-picture sentence structure is quite familiar to an English speaker.

A simple transliteration is:

Rofe hamishpakha sheli amar shehakhisun hazeh khashuv bimyukhad bakhoref.

A few pronunciation notes:

• רופא = ro-FE
• משפחה = mishpa-KHA
• חיסון = khi-SUN
• חשוב = kha-SHUV
• בחורף = ba-KHO-ref

The letter ח is pronounced with a throaty sound, like the ch in German Bach or Scottish loch.