Type of car rental

I don't think it matters much, as long as you have the size comfortable for you and your stuff. We drove an automatic all over the area and it was fine. I'd definitely go above a compact for three people.
We had an Audi and boy was it fantastic.
AutoEurope will give you a comparison across the companies.

great-thanks for the tip to help compare companies. As for pick up time....if flight arrives ATH 2:20 what is the estimated time for car pick up you think? We will not have checked bags.

2:45 should be okay

With 3 adult passengers plus luggage, even cabins, try not to take category A cars (Toyota Aygo, Fiat Punto or equivalent), they are really very small cars, even by European standards. Try to take a more powerfull car.
After a few hours of driving on the winding semi-mountain roads in the Peloponnese you may regret the few euros you saved.
The Audi that Valadelphia rented, if it's a A3 or Q2 model, is probably almost double the price of the Cat A small/mini cars.

But it's up to you.

You might want to check ACR Athens Car Rental. The have a great reputation and will deliver the car to you at the airport arrivals rental, They monitor the status of your flight and meet you at arrivals door. All the paperwork is done on the hood of the car and away you go. One tip they do a walk around the car to identify any dings or scratches, Its a good idea to take a video of the car during the walk around. I missed a ding but my wifes video showed it during the walk around. You should do that with any rental car,

By the way nice choices of destination.
Here are some images of Nafplio. On the way to Nafplio you might want to stop at Corinth to see the canal and then stop at Ancient Corinth and right next door the amazing fortress at Acrocorinth. It has 360 degree views over the countryside. Well worth doing. Also consider a stop at Ancient Nemia. It is about 5 KM off the road where you turn south to Nafplio. It was a sister city to Olympia and had similar games. The stadium is still there.

Some images
Nafplio https://www.flickr.com/photos/stanbr54/albums/72157632094108982
Delphi and Meteora https://www.flickr.com/photos/stanbr54/albums/72157698930270992

Corinth Area. https://www.flickr.com/photos/stanbr54/albums/72157711755443483

Rental car is asking for contact number. Do you just give them your US Cell number?