Yazlık gömlekleri ütüledim, kışlık çizmeleri ise kutuya kaldırdım.

They are adjectives made from nouns:

• yaz = summer
• kış = winter
• -lık / -lik / -luk / -lük = a suffix that can mean for, related to, or suitable for

So:

• yazlık gömlek = a summer shirt / a shirt for summer
• kışlık çizme = a winter boot / boots for winter

In this sentence, yazlık and kışlık describe the type of clothing, not ownership.

Here, -leri is the combination of:

• -ler / -lar = plural
• -(y)i / -(y)ı / -(y)u / -(y)ü = accusative case

So:

• gömlek = shirt
• gömlekler = shirts
• gömlekleri = the shirts

And:

• çizme = boot
• çizmeler = boots
• çizmeleri = the boots

The accusative is used because these are specific, definite direct objects: not just any shirts or boots, but the summer shirts and the winter boots.

Turkish often marks a direct object with the accusative when it is specific or definite.

So:

• gömlekleri ütüledim = I ironed the shirts / those shirts
• çizmeleri kaldırdım = I put away the boots / those boots

If the objects were indefinite, Turkish would usually avoid the accusative. So the case marking helps show that the speaker has particular shirts and boots in mind.

Ütüledim breaks down like this:

• ütüle- = to iron
• -di = past tense
• -m = I

So ütüledim means I ironed.

Turkish usually includes the subject inside the verb ending, so there is no need to say ben unless you want emphasis.

Not necessarily. Kaldırmak has several meanings depending on context, including:

• to lift
• to remove
• to clear away
• to put away / store

In this sentence, because of kutuya (into the box), kaldırdım means something like:

• I put them away in the box
• I stored them in the box

So this is a very natural household/clothing context, not just physical lifting.

Because -ya here is the dative case, which often means to or into.

• kutu = box
• kutuya = to the box / into the box

With a verb like kaldırmak in this context, the dative shows the destination:

• kışlık çizmeleri kutuya kaldırdım = I put the winter boots into the box

Also notice the y: it is a buffer consonant added between two vowels.

İse adds contrast or topic shift. Here it works like:

• as for
• whereas
• on the other hand

So the sentence contrasts two actions:

• I ironed the summer shirts,
• but/as for the winter boots, I put them away in the box.

It helps show that the speaker did one thing with one group of clothes and a different thing with another group.

Turkish normally puts the verb at the end of the clause.

So instead of English-style:

• I ironed the summer shirts

Turkish prefers:

• Yazlık gömlekleri ütüledim
• literally: summer shirts-the ironed-I

The same happens in the second clause:

• kışlık çizmeleri ise kutuya kaldırdım
• literally: winter boots-the as for into-the-box put-away-I

This is standard Turkish word order, not poetic or unusual.

Because the verb already tells you the subject.

• ütüledim = I ironed
• kaldırdım = I put away

The -m ending shows first person singular. Turkish often drops subject pronouns when they are clear from the verb. Saying ben is possible, but it would add emphasis, like I ironed them.

Yes, forms like gömlekleri can sometimes be ambiguous in Turkish. Depending on context, it can mean things like:

• the shirts
• his/her shirts
• their shirts

But in this sentence, the most natural reading is the summer shirts and the winter boots as definite direct objects, because:

• they are paired in a list,
• both are followed by past-tense verbs,
• no possessor is mentioned.

So although the form can be ambiguous in isolation, the sentence context makes the intended meaning clear.