Nafplio, Milos & Athens suggestions needed please

I understand not liking to drive, but I would consider how you will get around in the Nafplio region without one. A rental car would also likely be cheaper than paying for private transfers. There is a bus to Nafplio though.
Next steps: I would check flight schedules to Milos (task 1: Is there an airport?) and see if you could get there on arrival evening, stay X days, then return to ATH, rent a car for a few days in Nafplio, then return to wrap up in Athens.
It’s not ideal for 9 days, so you could also consider sticking to the Peloponnese, where you will find no shortage of beautiful beaches.

Let me tell you about a lovely walk at Nafplio that locals often do. If you are facing out at the marina, go to your far left until you get to a gate that makes it look like the trail is closed. It really isn't. Just scoot around the gate and enjoy the lovely paved path around the headland with amazing views. It ends on the other side of the headland where there is another gate and another obvious path around it. There is a swimming beach at this location or you can backtrack or walk up the road back to the center of town.

You could take a flight to Milos as soon as you arrive in Athens.
In September you will only have two options: the Olympic flight at 17:55 and the Sky Express flight at 18:45.

That said I don't know why you chose Milos, it's maybe because you saw Sarakiniko beach on photos or videos that promoted the island as an Instagram Island.

About the beaches, if it should be an important part of your activities, just to warn you that if you don't rent a car the bus network is not convenient to get to the few beaches of Milos.
You can basically go to the village of Pollonia in the north of the island where there is a beach, to Provatas beach in the south and to the famous Sarakiniko beach. That's it.

Anyway Milos doesn't have many beaches easily accessible by road. A large part of the island is uninhabited, there are only 3 villages on Milos which explains why the road network is very sparse, the other roads are dirt roads impassable by a standard vehicle.

valadelpia has already explained why renting a car to go to Nafplio would be the best option in terms of cost and time.
A private driver should cost around 200€ to 250€ one way (the equivalent of a week's car rental)

A round trip Athens-Nafplio by public bus costs 26€

Thanks everyone for the info and suggestions! I figured we will need a car on the island but did read that a lot of Nafplio is walkable and I'm perfectly fine taking taxis whenever needed so I appreciate the bus info and will look further into everything.

I don't really care about Instagram:) but Milos does look beautiful and unique. I've also read so many positive reviews of the island and it seemed rather doable to have a few different experiences there in just the few days we will have there, but definitely open to other suggestions.

It actually does work with the flight times (thanks JoLui!) to get to Milos on arrival evening, so if that is what you want to do, it works! I would choose to maximize my time and do less moving around, but that is of course not the only way to go about it.
What did you want to do FROM Nafplio? Most want to explore the sites in the region, and getting to Epidaurus, Mycene is not super easy, plus with a car you can stop en route at Corinth, Nemea. If you absolutely don't drive, state that clearly, but it's a highway to get there and then fairly rural roads in the area. Nafplio itself does not require a car but with a few days I would imagine you would want to explore. Hope that helps.
Write it all out day by day with travel times to see if this plans gives you enough time for what you want out of the trip. Good luck!

@valadelphia - sorry for the delay. Thanks for the tips. Yeah we definitely don't want to rent a car. I hate driving in my own city and want to fully relax and enjoy myself when on vacation - and for us that means not driving - even if it complicates things it is what has worked for us in the past. Just looked at those flights - thanks. We may be switching to Naxos (vs Milos) due to being able to navigate it more easily without a car or so many taxis/tours. I think when we get to Nafplio we may stay put more in town and just explore locally, than visit the other sites. I know 3 locations may be a lot for 10 nights, but at the same time we tend to stick to pretty relaxed itineraries in the places we visit, and just try to get more of a feel of things rather than take more day trips and squeeze more things in. Appreciate the suggestions!