How to plan Athens sightseeing when flying in and out of Athens

I'm wondering if we should plan to visit Athens at the beginning or
end of the trip.

Indeed, there are pros and cons. The argument for saving Athens for the end of the trip is that it's better to recover from jet lag on an island than in Athens, especially if you're going to an island with an airport first. I can't comment on that since I have a 2.5hour flight to Athens with no jet lag.

The other point is that if you're visiting Athens at the end of your trip (after the islands), being there means you don't risk a cancellation, strike, or any other event that would delay your return flight or ferry to Athens and cause you to miss your flight back to Canada.

I can completely agree with that, having experienced this kind of inconvenience twice in my carefree early travels in Greece when I was stuck on an island while my return flight took off from Athens without me. :)

You haven't specified the exact dates of your trip, but be aware that if you skip Santorini, you won't have many ferry options to get from Milos in the Western Cyclades to Naxos. It will either mean making a stopover in Santorini anyway, or taking a small and uncomfortable Seajets catamaran that will take an endless journey via several islands before dropping you off in Naxos.

There is also a conventional weekly ferry, which also makes quite a few stops before arriving in Naxos.

The number of days you want to stay in Athens depends entirely on what you want to do/see. Athens has enough to keep you busy for a whole week, but let's say a first proper visit requires 3 full days.

Personally, I always keep at least 4 days in Athens at the end of my trip, I've done this for years and still haven't finished exploring this city.

Thank you. This is very helpful. I will look at Athens for 3.5 days at the end of the trip and just one night, day of arrival, at the beginning now. Would you know if it is better to go from Athens to Milos, Santorini, Naxos, Athens or Athens Naxos, Santorini, Milos, Athens? I am consulting the ferry schedule but it is always good to hear what others know.

In terms of daily ferry availability, both options are good. You'll need to check which one suits you best based on the day's ferry schedules.

The main thing, from a ferry route perspective, is to plan to stay in Santorini between Naxos and Milos, regardless of the direction.
If you're not taking a connecting flight to the first island, you can spend the first night in a hotel in the port of Piraeus, so you'll be there to catch the first ferry in the morning to Milos or Naxos.

From the airport, you can go directly to the port of Piraeus by metro (€9) or by bus X96 (€5.50). There are some inexpensive hotels in Piraeus located near the ferry port docks. Keep in mind, however, that the port is huge and departures are from numbered gates (1 to 10), depending on the destination and the ferry company.

See detailed explanations here:

https://www.ferryhopper.com/en/destinations/greece/piraeus

On this webpage, in the section "From Athens airport to Piraeus port," the updated information for metro line 3, which now runs directly from the airport to the port of Piraeus, is missing.
The prices are also the ones I provided, not those mentioned on this webpage.

Since you’re arriving early afternoon, see if you can get a flight to Naxos or Milos within a couple of hours of landing. It’ll give you almost a whole extra island day — compared to taking the metro to Piraeus, spending overnight at a hotel and a taking a morning ferry. We have a long flight too, and have tried both methods (Athens first or direct to an island) and much prefer just pushing through and getting there.

We do always make sure we have transportation to the island hotel arranged in advance — so we can step off the plane, see someone with our name on a sign, and relax.

I have done what Charlene is suggesting twice and it certainly is preferable to “wasting” a night near the airport or port.