Which places of Greece should you visit?
Athens: You would need to spend at least two days in the historical centre, where you can visit all the places of historical and archaeological interest such as temples, ruins, museums etc. In Plaka you will walk in narrow roads where you can enjoy the typical and traditional Greek architecture and in Monastiraki you will do your shopping and taste the best souvlaki and tzatziki in Greece!
Disadvantage: The area of Athens is not famous for its beaches so I don't advise you to visit Athens in the summer if you are not planning to visit the islands as well. In my opinion, the only beach that worths visiting is down to Cape Sounio, where you can see the Temple of Poseidon and enjoy an amazing sunset, but this is more that 1 hour far from the city of Athens.
Thessaloniki: It is the second biggest city with its own airport in the North of Greece. Thessaloniki is different than Athens as it's a typical Mediterranean city next to the sea (like Nice, Barcelona or Napoli) also it feels a little bit like Istanbul and there is a lot of Byzantine culture (museums, churches, castles). If you have planned to visit Thessaloniki in the summer, then you will need another flight to the islands, or you will visit the beaches of Chalkidiki by bus (1 or 1,5 hour by bus), where you can swim in some of the most beautiful beaches in Greece.
Advise: Athens and Thessaloniki can get very hot in July and August and this would make you suffer during your sightseeing. April, May, June and September or October are the ideal months to visit those cities.
Mykonos and Santorini: Both islands worth visiting, however you need to take into consideration that during summer they get very crowded and everything is ridiculously expensive. If you are a young person and don't mind paying the money, then partying in Mykonos in July is probably a priority for you so I guess you will pay the money in the end, but if you need an 'alternative' experience, you'd rather visit those islands in middle September. The weather is still warm and everything is half price and quieter. Another good idea would be to stay on one of the islands that are close to Mykonos and Santorini (you would pay half the price) and pay visits to those islands. All the islands in the Cyclades (is the name of the islands that surround Mykonos and Santorini) worth visiting for their traditional white and blue houses, food, beaches and way of life.
Meteora: Famous rocks with monasteries on the top. It is an UNESCO World Heritage Site and worths visiting. The only disadvantage is that it is far from Athens of Thessaloniki, it is in the middle of nowhere in a way and thus, you will have to do a 3 or 4-hour trip from Athens or Thessaloniki and it is even more difficult to get there if you are on one of the islands.
Corfu, Rhodes: Famous islands for their medieval architecture, beaches, nightlife and more.
Crete: The biggest island of Greece. If you need a combination of mountains and rocky or sandy beaches, big cities or small towns and villages, history and culture, amazing food, then Crete has everything. Due to the fact that it is so big, Crete can satisfy every need. However, in order to be able to explore the whole island, you would need one month. If you are going to visit Crete only for 5 days, then just visit the biggest cities and the most famous beaches or only the places that are important to you.
Green islands: On those islands your experience is different. as they don't belong to the Cyclades where you can see the wild beauty of the dry hills and the typical Greek architecture.
These islands are: Skiathos (is also called Mykonos of the North as recently has become a very popular island with its own airport), Skopelos (famous from 'Mamma Mia' movie), Alonissos (a very alternative experience for people who love nature), Ikaria (famous for the immortality of the people who live there), Thasos (pine trees and exotic beaches), Zakynthos (famous for 'Nauagio' beach), Leykada (famous for the green-blue colour of the sea), Kefallonia (famous for 'Myrtos' beach),
Less known islands that worth visiting: Samothrace and Kithera (sophisticated and famous for their springs and quiet environment), Paxoi (if you are in Corfu, it's easy to spend one day there), Samos (the island of Pythagoras), a lot of little islands in the Cyclades or the Dodecannese.
Towns in mainland Greece: Nafplio (in my opinion, the most beautiful town in Greece), Kalamata (famous for its olives), Kavala (close to Thasos island), Volos (close to amazing beaches in the area of Pilio mountain), Giannena (interesting city but unfortunately far from the sea), Nafpaktos (very beautiful town which is also close to the Oracle of Delphi), Kastoria (a beautiful town with a lake but unfortunately very far from the sea).
For a full Greek experience you would need at least 15 days. Spend three days in Athens, one day in Delphi, two days in Nafplio and ancient Epidauros, two days in Mykonos, two days in Santorini, two days in Thessaloniki and three days in Crete. This obviously includes a lot of travelling by plane, boat, bus, train or car but will give you a taste of everything!
Another idea would be to visit Athens for a couple of days and then drive West or South - West down to the Peloponnese. You can visit Nafpaktos and Delphi, Nafplio and Epidavros, Ancient Olympia, Kalamata and Monemvasia, and of course if you have more days, little islands such as Ydra, Poros and Kythira. For this trip you would need at least 10 days and ideally you would do this by car so you can stop wherever you like!
I recommend this trip because it includes everything: the most important archaeological sites, a lot of beaches on your way, beautiful traditional towns and villages etc. Well, it doesn't include the islands that everybody wants to go to, but if you have 5 mores days, you can go to the islands as well or you can go to the islands during your next visit to Greece. Or you can visit the islands of Kythera, Poros, Ydra and Spetses which are very easily accessible from the Peloponnese.

Advise: Whichever country you visit, you need to avoid touristic and crowded places in order to taste authentic and cheap food and meet authentic people. The same applies to Greece! But this is up to you, it depends on what kind of experience you'd like. It's just very common in some islands. Tourists arrive and they have prepaid for their stay and meals at the hotel. This makes you feel comfortable of course, you can sort out your expenses, but on the other hand it doesn't give you the opportunity to try other things. Well, if for example you want to try real Greek food and I mean the quantity and quality as well, visit the most isolated restaurant (taverna) in the most isolated village!
I hope this is helpful.
Feel free to email me if you need more information.
You can also do your own research. It's apparent that if you want to explore Greece, you need a lifetime so choose what is best for you. It's always up to you what kind of holidays you want to spend, noisy or quiet, cheap or expensive, how many days you are going to spend in Greece, if you prefer doing lots of travelling or staying in one place etc etc.
