We loved Greece as well. For our excursions:
- Rome - Day 1 - Slept once we got to the room as we were dead tired from the flight.
That's the worst thing that you can do for jet lag! It simply prolonged the jet lag.
Yes, it is hard to stay awake on the first day but giving in to your body only make it worse.
I hope your next trip is better with this tip.
-Paul