Learn Russian: Дим, я прошу тебя, чтобы ты перезвонил бабушке. - Dim, I am asking you to call grandma back.

Дим is the colloquial truncated vocative form of the name Дима. In informal spoken Russian, given names and kinship terms ending in -а or -я often drop this final vowel when you are directly addressing the person. It makes the speech sound more natural, fast, and intimate.

Could I just use the infinitive перезвонить here instead of чтобы ты перезвонил?

Yes, you absolutely can! Я прошу тебя перезвонить бабушке is very common and concise. Using чтобы plus the past tense means the exact same thing but adds a slight bit of emphasis to the request, clearly separating your action of asking from his action of calling back.

What does the prefix пере- add to the verb перезвонил?

The prefix пере- often adds the meaning of doing something "back" or "again" to a verb. When added to звонить (to call), it creates the perfective verb перезвонить, which specifically means "to call back" (like returning a missed call) or "to call again later".

Why is бабушка in the dative case (бабушке)?

The verb перезвонить (just like the base verb звонить) requires the person being called to take the dative case. Think of it as directing the action toward them, essentially making a call "to" grandma.