Experienced travelers to Greece recommend that you get out to your first island immediately on Arrival in Greece. Getting over jet lag on an island is way better than doing it in Athens. Athens is very busy and is a culture shock for first timers.
If I were doing your trip I would tough it out at Athens airport. We have done that on late arrivals. Here is some info.
Amenities at Athens Airport are simple and to the point. Several restaurants are open 24-hours, and there are art installations and a museum to keep you entertained briefly at least. WiFi is available to pass the time. For those looking for a little added comfort, there are first class lounges that economy class passengers can pay to access.
There are early morning flight to Santorini I believe around 6am. Crashing in Santorini and waking up in the afternoon to stunning caldera views is magical. Spend you three nights in Santorini. Ferry to Naxos. I believe there is an early morning ferry the Blue Star which leaves around 6am. I can't actually find that on the Greek ferry schedule but there are two fast ferries that do the Naxos run golden Star and seajets which depart between 9 and 10 am. Both will get you into Naxos in less than 2 hours.
Three nights in Naxos is the bare minimum and you will basically have time to explore Naxos town and perhaps a bus ride up to one of the lovely mountain villages.
Stay at St George beach. It is part of Naxos town so if the sun is out you can get some beach time and still be in Naxos town after a 10 minute stroll along the seaside.
Now ferry or fly back to Athens. Rent a couple of cars at the airport or a van and drive to Corinth. Renting at the airport means you do not have to drive through Athens. Use the toll road National Highway.
I doubt you will be able to visit Corinth area including Nemia, ancient Corinth, Acrocorinth and still have time and energy to do Nafplio and Epidavros. Drop Aegina you will already have had your island experience in Naxos and Santorini. Use that day to stay overnight in Nafplio. You won't regret missing Aegina, you will regret missing Nafplio.
By now you will be acclimatized to Greece and will be prepared for your Athens experience. Do Athens last.
Here is what you will see.
Naxos Town http://www.flickr.com/photos/stanbr54/sets/72157632094558042/
Trip around Naxos http://www.flickr.com/photos/stanbr54/sets/72157634605629689/
Nafplio and Peloponnese http://www.flickr.com/photos/stanbr54/sets/72157632094108982/
Corinth area sights https://www.flickr.com/photos/stanbr54/albums/72157711755443483
Other thoughts The day trip from Naxos to Delos and Mykonos may not be running in April. stop in at one of the travel agencies on the Naxos waterfront Promenade and see if the boats are running. This early in the season they may be taking names of people interested in the excursion and when they have enough people on the waiting list they might put a ship on that run. I expect this trip might not work for you. Besides there is more than enough for you to do on Naxos for a week.
Have a great time.