Best summer month to visit Greece is in July. What's the difference you might ask? Well I'll tell you. Greece has A LOT of tourism all year around. It has and continues to be a popular vacay destination for its blue skies and seas. Traditionally in the month of August is when all of the Greeks take their mandatory vacation time and so they are also traveling domestically (for them) to islands and beachside resorts. In my experience, Greece is better in July. You will find more availability in hotels, car rentals, reservations for restaurants, etc. It is also less hot in July. I don't know why that is, but it has always been my experience. Also the day is shorter in August. In July the sun goes down at 8:30 and you can enjoy more activities and sunlight. In August (especially the end of) the sun starts to set at around 6:30 pm. I am not a fan of less time at the beach. I have been to Greece in June, July, August and September. And my own personal recommendation to enjoy the beaches and water is to visit in May, June and July. Clearly temperatures vary and I am not a weather woman, but this is my experience and advice. Lesson: skip Greece in August.
Once your time of year to visit is settled, you are definitely going to want to pick an island or two to visit while there. The islands are what bring most visitors to Greece. Its the climate, lifestyle, scenery and cuisine unmatched in any other county in the world. You can chose a party island like Mykonos, where you will sleep all day and dance all night or quiet islands Naxos or Lefkada. I recommend traveling to major islands by plane in the summer. If you are on a budget then the ferry boats are your best bet. I've traveled both and they both work. Once on the islands, you will need transportation. I definitely recommend either choosing between a rental car or moped to get around. Most major islands like Mykonos and Santorini have buses but that will take more time. And the smaller ones you definitely need a mode of transportation to explore on your own.