Learn Hardcore Greek: Ρωτάω συχνά τη γιαγιά μου για την ηλικία της και για την παιδική της ζωή. - I often ask my grandmother about her age and about her childhood.

ρωτάω means I ask (a question).
ρωτώ is essentially the same verb; it is a slightly more formal or slightly older-sounding variant. In everyday speech, ρωτάω is more common.

Do not confuse ρωτάω (ask a question) with ζητάω (ask for / request).

Ρωτάω τη γιαγιά μου = I ask my grandmother (questions).
Ζητάω από τη γιαγιά μου χρήματα = I ask my grandmother for money.

Why is there no word for I (like εγώ) in the Greek sentence?

Greek usually drops subject pronouns because the verb ending already shows the subject.
Ρωτάω is first person singular, so it already means I ask.

You use εγώ mainly for emphasis or contrast:

Εγώ ρωτάω συχνά τη γιαγιά μου… = I (as opposed to someone else) often ask my grandmother…

Why is it τη γιαγιά μου and not something like στη γιαγιά μου?

The verb ρωτάω takes a direct object in the accusative case, without a preposition:

ρωτάω κάποιον κάτι = I ask someone something.

So τη γιαγιά μου is accusative: (εγώ) ρωτάω τη γιαγιά μου.
Using στη γιαγιά μου would mean to my grandmother and would be wrong with ρωτάω in standard Greek.