Ağustos tatilinde güneşten korunmak için şapka takmalıymışız.

A natural breakdown is:

• Ağustos tatilinde = during the August holiday / on the August vacation
• güneşten = from the sun
• korunmak için = in order to protect oneself
• şapka = hat
• takmalıymışız = apparently we should wear / we were apparently supposed to wear

So the structure is basically:

[time] + [purpose] + [object] + [verb]

That is very common in Turkish.

Ağustos tatilinde means during the August holiday/vacation.

It is more specific than Ağustos'ta, which just means in August.

• Ağustos'ta = in August
• Ağustos tatilinde = during the August vacation/holiday

So the sentence is not talking about the whole month in general, but specifically about the holiday period in August.

This comes from a noun phrase plus the locative ending.

First, Turkish forms a compound-like structure:

• Ağustos tatili = August holiday

Here, tatili has a possessive-type ending because that is how Turkish commonly builds noun compounds.

Then you add -de/-da to mean in/on/during:

• tatili + nde → tatilinde

So Ağustos tatilinde literally works like in its August holiday, but in natural English it is just during the August holiday.

Because the verb korunmak usually takes the thing you are protecting yourself from in the ablative case:

• -den / -dan / -ten / -tan = from

So:

• güneşten korunmak = to protect oneself from the sun

This is a pattern you should learn as a unit:

• soğuktan korunmak = protect oneself from the cold
• yağmurdan korunmak = protect oneself from the rain
• güneşten korunmak = protect oneself from the sun

This is an important one.

• korumak = to protect something/someone
• korunmak = to protect oneself / to be protected

Examples:

• Çocuğu güneşten korumak = to protect the child from the sun
• Güneşten korunmak = to protect oneself from the sun

In your sentence, the meaning is reflexive: we should protect ourselves from the sun, so korunmak is the right form.

için means for or in order to.

Here it links the purpose to the main action:

• güneşten korunmak için şapka takmak = to wear a hat in order to protect oneself from the sun

A very common Turkish pattern is:

verb stem + -mak/-mek + için

Examples:

• dinlenmek için = in order to rest
• öğrenmek için = in order to learn
• korunmak için = in order to protect oneself

Both can relate to wearing, but they are used differently.

• giymek is used for clothes you put on your body, like shirts, pants, coats, dresses.
• takmak is used for things you attach/put on such as hats, glasses, jewelry, watches, masks, and similar items.

So:

• gömlek giymek = to wear a shirt
• pantolon giymek = to wear trousers
• şapka takmak = to wear a hat
• gözlük takmak = to wear glasses

That is why şapka takmak sounds natural here.

-malı / -meli expresses necessity, obligation, or strong advice.

So:

• takmalı = should wear / must wear depending on context

Examples:

• Gitmeliyim = I should go / I must go
• Daha çok su içmelisin = You should drink more water
• Şapka takmalıyız = We should wear hats

In your sentence, this is the core idea: wearing a hat is necessary/advisable.

This is one of the most interesting parts of the sentence.

• takmalıyız = we should wear
• takmalıymışız = apparently we should wear / it turns out we should wear / we were supposed to wear, it seems

The ending related to -mış often adds meanings like:

• reported information: they say
• realization after the fact: it turns out
• indirectness: apparently

So takmalıymışız is less direct than takmalıyız.

Possible nuances:

• Someone told us this recommendation.
• We have just found this out.
• Looking back, we realize that we should have been wearing hats.

So this form is not just plain advice; it carries an extra sense of report or discovery.

It can be understood as:

• tak- = wear/put on
• -malı = should / must
• ymış = apparently / it seems / reportedly
• -ız = we

So:

tak + malı + ymış + ız → takmalıymışız

The y is a linking sound that helps pronunciation between vowels.

This kind of stack is very normal in Turkish: a verb can take several layers of meaning in one word.

Because Turkish usually shows the subject in the verb ending.

Here, -ız in takmalıymışız already means we.

So biz is not necessary.

• Şapka takmalıymışız. = We apparently should wear hats.
• Biz şapka takmalıymışız. = same meaning, but biz adds emphasis, contrast, or clarity

Turkish often drops subject pronouns unless they are needed for emphasis.

Not completely. Turkish word order is flexible, but the version you have is very natural.

The standard neutral order is:

Ağustos tatilinde güneşten korunmak için şapka takmalıymışız.

That places the main verb at the end, which is very typical in Turkish.

You could move parts around for emphasis, for example:

• Güneşten korunmak için Ağustos tatilinde şapka takmalıymışız.
• Şapka takmalıymışız, Ağustos tatilinde güneşten korunmak için.

But the original sentence sounds the most neutral and natural for ordinary use.

You would usually say:

Ağustos tatilinde güneşten korunmak için şapka takmalıyız.

That means simply:

We should wear hats during the August holiday to protect ourselves from the sun.

So the contrast is:

• takmalıyız = direct advice/necessity
• takmalıymışız = reported or newly realized advice/necessity