Just starting to plan a trip!

You are on the right track--thinking about how you wish to spend your days. It took me forever to get to Greece, as I found the choices overwhelming. After a few driving trips in Italy, it became clear that exploring the Peloponnese would be the perfect first trip for us. The concentration of ancient sites and the many hiking opportunities cinched it. With two weeks, you can include some island time as well. Late May was wonderful so I am sure June will be great as well. Feel free to ask about hiking and such--it's always my focus because we like to be mostly outside in nature.
I'd start with a guide book. Also look at trip reports and itineraries--they usually show you what you don;t want to do as much as what you do want to do.

You might also like to first consider whether you want to stay on the mainland and visit the islands, or do it in reverse. Or a mix. Obviously each would lend itself to a very different trip!

We stayed on an island (Aegina) and took day trips to Athens and to other islands.

We've found the handiest way to see the Greek islands is by cruise ship. Cheapest way, too. We also caught Malta and Turkey coming and going from Italy.

TripAdvisor offers a wealth of information, reviews, opinions, suggestions on things to do, etc. Each island has its own forum.
Booking.com is excellent for finding affordable accommodations.

A huge selection of islands to select from. They range from over-the-top touristy to more laid back and traditional. June will still be outside the High Season (July/Aug.) so you may be able to avoid the cruise ship crowds that flock to islands like Santorini or Mykonos.

I have my favorite Greek Islands but they may not be what you want. However I can recommend:

Amorgos
Tinos
Astypalea
Naxos
Crete
Chios
Sifnos
Milos
Ikaria (not recommended for a first time traveler to Greece, however) truly a different Greek Island but in a good way!

I'll be going to the island of Lesvos this October and it sounds perfect for me and you may like it also.

The mainland is more varied, especially the Peloponnese with a huge variety of towns, villages, archeological/historic sites, beaches and lots more.

I think more research on your part to find more in line of what you are looking for and then come back and ask about what you found.

A couple of very good Guide Books I can recommend are:
Lonely Planet (Greece)
Blue Guide to the Aegean Islands

Check out this wonderful website for Greece:
https://www.greecetravel.com/

I'll be on Chios and Milos for the first time this coming September and I'm looking forward to some new adventures.

We went May-June 2018 for two weeks. Advice I got and was happy with the results is to 1) fly to furthest island first from Athens after arriving 2) stay within one island group for ease of travel 3) leave days in Athens all at the end.

We flew to Athens and then on a separate flight a few hours later flew to Santorini. We stayed 3 nights. Then we ferried to Naxos and stayed 4 nights. Then we flew back to Athens, rented a car, and drove to Nafplio where we stayed 4 nights. We drove back to Athens, returned the car at the airport and stayed in Athens 3 nights.

We loved our trip and felt it was a good balance of islands and mainland attractions for first time visitors.

You were wise to give Italy the two weeks you had this year, and if you liked it you'll like Greece. This website has tons of information about travel in Greece. The "Explore Europe" link on this website has a lot on Athens and the mainland, as does the RS guidebook, but nothing on the islands. Other guidebooks cover both areas.

I've been to Crete but not the Aegean islands most often recommended. From what I've read I would avoid Mykonos and probably also Santorini because of their popularity. For a trip involving both mainland and islands (and two weeks would give you time for both), most people seem to recommend starting with islands to recover from jetlag and ending in Athens to minimize risk of missing the flight home.

A cruise among islands could be a great experience, I think MSC and other lines do them from Athens over multiple days. You could also put together some combination of ferries and flights. We loved Crete, but it's much bigger and takes longer to "see" than the smaller islands north of it.

Athens is worth several days if you're interested in the ancient civilization, with the Acropolis, two world-class museums, and much else. Nafplio is a beautiful seaside city and a good base for other ancient sites like Mycenae, Epidavros, and Nemea. Delphi is unique and worth an overnight.

Signage in Greece is mostly bilingual, and you could manage with the usual collection of polite handy phrases, but at my wife's insistence I learned the Greek alphabet. That opened doors for understanding my surroundings, since knowing how to pronounce words often gives you a leg-up in understanding their meaning.

I second the recommendation of the Lonely Planet guide for the Greek islands. I have just started researching 2 weeks in Greece in May 2020 and am finding it excellent. The real problem is narrowing it down.

Nicole, in addition to previous guidebook recommendations, we found "The Rough Guide to Greece" a comprehensive resource. Our favorite Greek Island for accomodations, beaches, swimming, sightseeing and cuisine was Naxos. Once you have come up with a preliminary plan come back! There is a large contingent who have been to Greece and the Greek Islands. We received advice from some of the Forum experts prior to our trip a couple years ago.

Hello,
This year me and my fiance decided to visit Greece. We visited Crete. I really recomend it because its easy to access (there are airports) and its a beautiful island. We have stayed in Falasarna. It was beautiful. The beach was amazing. There are no any clubs or anything like this but you can do water sports. If you are looking for a peaceful place i think this is it. We have also been to Alonnissos 2 years ago. It was a beautiful also but harder to access. We had to take a ferry from Salonicco. I have found a nice video showing Falasarna beach. I hope that once you see it you will like it.
https://www.youtube.com/watch?v=xR_Gtv6A81I&feature=youtu.be