Learn Turkish: Lütfen dergiyi okuyun. - Please read the magazine.

This is the accusative case in action! Because you are asking someone to read a specific magazine ("the magazine"), it is the definite direct object of the verb. We add the suffix -i to dergi, but since dergi ends in a vowel, we need the buffer letter y to keep the vowels apart, making it dergiyi.

If I wanted to say "Please read a magazine", would I still use dergiyi?

No, you wouldn't. The suffix -(y)i makes the word specific ("the magazine"). If you wanted to say "a magazine" (any magazine in general), you would drop the suffix and just say Lütfen bir dergi okuyun.

Why does the verb say okuyun instead of just oku?

Oku is the informal, singular command for "read". By adding the suffix -un (with a buffer y because oku ends in a vowel), it becomes formal or plural. Since we are using lütfen (please), it is natural to use the more polite, formal version of the verb.

I noticed both words use a buffer y (dergi-y-i and oku-y-un). Is this a rule?