Driving from Patras to Nafplio

Google Maps will show rental car agencies if it comes to that, but tacking onto your Italian thread--I think you'll most likely end up flying, and that would be so much easier. Rent car at ATH and drive to Nafplio. How many days? I am assuming the Rome thing is just a typo.

Stay in Nafplio and take day trips out to tour Rome

Day trips to Rome from Nafplio?

I guess you have made a typo

Lol. Yes, that's a typo! Day trips around Peloponnese!

We will be in Nafplio for 4 days.

If you fly round-trip to ATH and head to Nafplio, you could return the car late afternoon/early evening on day 3, spend the night in Athens, and have a full day to see what you wish to see. I would not cut it much more closely than that, though if your flight is really that late in the day, I suppose you could return from Nafplio in the morning, store luggage at airport, hop in a cab to the Acropolis, then return to airport.
One day was sufficient for us to see the Acropolis, walk around a good bit, and have a nice dinner. (We'll be passing back through Athens the next time we get to Greece most likely, so we did not feel pressure to see it all.)
I did not include Hydra in my plans, there is plenty to see close to Nafplio, and it complicates things on a short trip.

Picking up a car in Patras to drop off in another region (e.g. Athens) will incur additional costs and limit the choice of car rental agencies.

However I had a look and found at least one local rental company in Patras that allows it (and has automatic cars).
The one way rental charge is €120.00 for a drop off at Athens airport ( to be added to the price indicated).
(a major rental company like Enterprise that also has an office in Patras will charge twice as much for one way rental)

You can check their website for comparison with other rental companies:

https://theocar.com/en/car-rental/

In my opinion it is unlikely that you will find a rental company that allows you to pick up a car in Patras and drop it off at the port of Piraeus in order to catch a ferry. So returning it to Athens airport will probably be your best option.

Going to Hydra (why Hydra by the way?) or another Saronic island is also possible from Porto Heli which is only 1h30 from Nafplio (see here: https://maps.app.goo.gl/8CHJopVo4w1ohHG77)

Thank you so much for the reply. Unfortunately, I think flying a family of 4 is going to cost too much; from what I'm finding the ferry is cheaper... does anyone know what the best way to purchase a fight from Rome to Athens is?

You make a really good point about having to return the car to Athens... thank you for the link to the rental car agency!

We picked Hydra because of the lack of cars and the donkeys. We thought it would be a good cultural experience for the kids to see something so different.

does anyone know what the best way to purchase a fight from Rome to
Athens is?

Check on websites like Skyscanner or Googleflight to find flights Rome→Athens

Usually, 4 airlines operate direct flights Rome→Athens:
Ryanair - ITA - Sky Express - Aegean
The best choices are in my opinion Sky Express or Aegean, the two Greek airlines. Aegean is regularly ranked among the best European airlines.
Ryanair is a low-cost airline that offers low prices if you don't you have any baggage (not even cabin) and with which you will have to pay for the slightest glass of water on board.
ITA is the Italian national airline that I have not used recently and which has had some financial problems and should be bought by Lufthansa (it could happen today).

If you fly to Athens you could rent a car at the airport and drive straight to Nafplio. Then come back to Athens and return the car either at the airport or at the port of Piraeus to take a ferry to Hydra.

There are indeed donkeys and no cars in Hydra, but be aware that there are no sandy beaches either, only a few small pebble beaches.