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You'll be spending nine weeks in the MTC and learning a new language in that time. It's not a whole lot of time, and you won't speak Greek perfectly by the time you leave, but with the intensive training you'll receive, you'll learn quite a lot.
You enter the MTC on a Wednesday and are encouraged to speak "only" Greek from your first Monday there (of course, by that time, you won't know very much, so this translates to "Speak Greek as much as possible"). This is good advice. Practise speaking Greek as much as you can. For the first four weeks, the teachers will speak English; beginning in week 5, they are supposed to speak only Greek to you, except for necessary explanations. Try to support them, please! It's not easy for them, either, having to speak only Greek, if you insist on speaking English back or not understanding on purpose (genuinely not understanding is fine, though).
I encourage you to learn Principle 5 of the first missionary discussion quickly; it is about the Book of Mormon, and this way you will be able to explain what the Book of Mormon is to a Greek person; and it's also a useful quick message at a street board, for example.
Based on the notes I took in the MTC, here's a rough week-by-week overview of what you'll learn in the MTC.