"Put the glass on the table."
Translation:Βάλε το ποτήρι στο τραπέζι.
Επί του τραπεζίου/τραπεζιού. Επί requires genitive. It's sophisticated, though.
πάνω? As in, βάλε το ποτήρι πάνω στο τραπέζι?
It's optional. The σε (in στο) has a lot of meanings - on, in, at, by, etc - and this already conveys the sense of the glass being on the table. If you really wanted to emphasise that you wanted the glass on top of the table, sure, you could use the πάνω, but it's optional. A bit like how pronouns in constructions like εσύ περπατάς are usually unnecessary (apart from emphasis) because the conjugation of the verb already tells us who is carrying out the action.
You also come across this quite a bit with μέσα, for "in"; it's often optional due to a σε.