Where to start - Greece or Turkey?

I suppose you plan tio visit some parts of Southern Italy too. Otherwise you could fly from Rome to Athens and back for about 150 Euro (all in) with Aegeanair/www.flyairone.it. Note that your car will complicate the visit of the Greek Islands: you will have to book the ferries in advance and you will have to pay for the car. Neither in Greece nor in Turkey it is a problem to get good public transport. Buses between Patras (the ferry harbour where you arrive from Italy) and Athens (180 km) run every 30 minutes..... Between Izmir and Istanbul, there is at least 1 bus per hour ....

In June it is much warmer in Rhodos than in Istanbul. So do your trip anti clockwise.
The following sites are worth a visit; I don't mention the small greek Islands like Santorini, Mykonos, Naxos , Paros ....... becouse it would get rather complicate with a car. My list i too big; it's up to you to choose:

Rome - Formia (trips to Ponza and Ventotene Islands) - Naples (Capri, Ischia, Sorrento, Pompei) - Matera (UNESCO world heritage) - Alberobello (Trulli) - Bari or Brindis - Ferry - Patras - Olympia - Pilos (Methoni, Koroni) - Kalamata - Sparta (Mistras) - Argos (Nauplia, Epidauros, Tyrins, Mykene) - Korinthos - Loutraki - Delfi - Ossios Loukas - Athens - Ferry - Chania - Paleochora (Samaria Gorge) - Festos - Iraklio (Knossos) - Malia - Iearapetra (Chrissi Island) - Palekastro (Kato Zakros) - Vai - Sitia - Ferry - Rhodos - Ferry - Marmaris - Fethyie - Olympos - Kas (trips to the Islands) - Myra - Phaselis - Antalya - Pamukkale - Ephesos - Izmir - Pergamon - Bursa - Yalova - Ferry - Kartal - Istanbul - Edirne - Kavala (Philippi, Thassos Island) - Chalkidiki - Thessaloniki - Pella - Florina - Bitola - Ohrid -Sveti Naum - Elbasan - Tirana - Kruja - Durres - Fier (Apollonia, Berat) - Vlore - Dhermi - Saranda (Butrint, Ksamil Beach) - Gjirokaster - Joannina (Dodoni) - Igoumenitsa - Ferry - Bari or Ancona - Highway - Rome.

Thanks so much for your kind advice.
We lived in Italy for quite a few years so we did all that. I guess traveling around Greek islands we don't need car except the main land and Crete but we plan to travel long distance in Turkey and prefer our lease car even if we need to take separate insurance for Turkey on top of what is generally provided for the lease car provider.
I will go over your list and maybe I can convince my husband we spend the whole two and a half month in Greece and leave Turkey until next year. This way we don't need the car as we can fly direct from Bangkok to Athens and back.

Your idea of spending the whole time in Greece is the best, you can rent a car for mainland and Crete and you can use the ferries for island hopping. From Rhodes you can visit Turkey if you want.

I wouldn't try to use the same car in Turkey. Too much hassle if you have any problems with it. Either leave out Turkey for this trip or rent a car separately in Turkey.

You could certainly see a lot of Greece in 10 weeks. There is also certainly a lot to see. I'd be tempted to stick to Greece and forget about a lease car altogther. You can rent a car fairly cheaply on most islands. That would eliminte hassle with the ferries.

See my reply to gemdrops on her post 'island hopping' on this branch for what to see and do on Rhodes as an example. You could spend 2 weeks doing that alone.

The ferry costs are exorbitant for cars on the Greek islands. It works out much cheaper to hire locally for a few days rather than bring your own transport. In your case you would have the expense for your leased car, plus the transport costs for the car on the ferries.
I hope this information helps to put some things in perspective regarding your choice.

Regarding your time frame, 2 + months is more than enough to enjoy Greece and Turkey, since this was your initial wish. You can easily make your way to Turkey from the following Greek islands -
Kos, Samos, Chios and Rhodes, or you can do the following-
Athens, pick up a rental car [one way, dropping off in Thessaloniki] visiting northern Greece - take the train to Istanbul, pick up a rental car [one way], and work your way down, cross over one of the above Greek islands[ drop off your rental car first], and visit the islands before working your way back to Athens, for your flight to Bangkok.
One way rentals can have surcharges, but they are usually very limited when drop off point is a main town/city

Regarding the weather, June is more comfortable regarding weather/price/crowds than July for Greece.

Thanks all for your time and advice.
My hubby who loves leasing car finally came to his sense and agrees that for once we will backpack (his term of traveling by public transport)through Greek islands and rent the cars as necessary. We are now seriously thinking of following dianab1's suggestion of flying in and out of Athens as we can use our mileage flying Thai Airways.
Crete is my favourite island and am excited to get back there again.