"Greece is prettier than you would imagine" - We had a wonderful time in our short visit (7 days) during May-June 2014. Since we had limited number of days, we chose the more popular destinations - Athens, Mykonos and Santorini. May - Aug is the peak season, but the place was relatively non-crowded (not over crowded) during May end-June beginning.
Athens is well connected to most cities in the world. It is a historical city and is oozes of rich Greek mythology and culture. The city is built around the famous Acropolis and is well connected by metro trains. Ferries from Athens can take you to most of the islands in Greece. Food was a bit of a problem for us in Athens - we did not find many decent restaurants close to where we stayed, although there were restaurants in the many hotels in the locality.
Mykonos is the prettiest town ever ! The tranquility of the sea, the beautiful clear waters, the flawlessly white streets in little Venice and the amazingly pretty resorts will mesmerize you to the core. Surprisingly, Mykonos is also a party town. There are quite a few parties happening all round the year at the popular beaches. A heaven for scuba divers, not only the regulars but also the beginners, this island has the best dives.
Santorini is the most popular and commercial islands of Greece. With an unmatched view of the Caldera, Santorini is easily one of the most beautiful spots for travelers. Everything is Santorini is beautiful - the cobbled pathways through the towns, the white and blue structures, the breathtaking view of the cliff and the sea, the many churches with huge bells, the tiny shops in Oia town, the chic bars and restaurants and much more. The icing on the cake - the private island tour on a boat, a 5 hour ship ride on the Aegean sea with a barbecue dinner and a view of the mesmerizing sunset - you wouldn't ask for more.
NOTE: it costed us more since we were on our honeymoon and chose to stay at resorts. You could easily reduce the amount by choosing the many pretty BnBs in Santorini and Mykonos.
Also, leave enough time to laze around and soak in the beauty of Greece apart from doing all the touristy activities.