Learn Turkish: Yaz için iki şort satın aldım. - I bought two pairs of shorts for the summer.

In Turkish, when you use a number before a noun, the noun always stays in its singular form. Since you already specified iki (two), adding a plural ending is not allowed. Also notice that while English says "pairs of shorts", Turkish simply says iki şort.

What is the difference between saying satın aldım and just aldım here?

Almak is a very common verb that can mean "to take", "to get", or "to buy". Satın almak specifically means "to buy" or "to purchase". In everyday speech, people often just say aldım when buying things, but satın aldım leaves no doubt that a purchase was made.

How would I use the hem... hem... structure from this lesson to say I bought shorts for both summer and winter?

You would place the conjunctions right before the two things you are linking. To say "for both summer and winter", it would become: Hem yaz hem kış için iki şort satın aldım.

Why is there no word for "the" in Yaz için (for the summer)?

Turkish doesn't have a separate word for "the". When talking about a season for a general purpose, you just use the bare noun yaz followed directly by the postposition için (for).