We did a similar trip a few years ago. Using Lee's good advice to save Athens & the mainland to the end of your trip, I'd recommend flying into Athens and then directly out to your furthest island, which is Rhodes. You'll be there in an hour vs. a 12 hour ferry ride. Allow 2 hours between your flight arriving in Athens and the flight out to Rhodes. 
Spend nights 1-2-3 in Rhodes. We stayed at the Attiki Hotel in Old Town and liked it very much. Ask the hotel to send a taxi to pick you up at the airport. Use your first night to recover from all those flights, the second to walk around Rhodes, the third to take a day trip -- I suggest the beautiful island of Symi. 
Day 4 &5  -- ferry to Leros. We loved quiet, laid-back Leros ... had some of our best food in Greece there. It has two main beaches -- Alinda & Xerocambos. We stayed near Xerocambos, at Villa Alexandros apartments, which were very nice, but isolated -- I think we were the only guests, and no one-spoke English!  Xerocambos had one of the most gorgeous bays we've ever seen, with sailboats anchored out, and it was almost deserted in late September. There was a taverna on the beach that had beach chairs in front you could use free if you bought food or drink. They offered standard Greek fare (souvlaki, grilled meats) but also "home cooking" (like chicken stroganoff or beef with pasta!) which were absolutely delicious. Be daring! For our last night on Leros, we stayed in a "real" hotel, the Crithoni Paradise -- with a pool, and breakfast.  And there was a great restaurant next door to the hotel called Esperithes Restaurant, with fabulous food that also did not sound Greek (like chicken stuffed with avocado!) but was wonderful.  Nearby was the immaculately-maintained Leros War Cemetery, with graves of 183 foreigners who fought the Germans there during WWII.  OR ... you can also stay in a windmill on Leros if you'd like. 
Day 6-7 -- ferry to Patmos. We stayed at the Blue Bay hotel, which was very nice. Patmos has St. John's Monastery and the Cave of the Apocalypse, where John is said to have written the book of Revelation. This is another quiet, lovely island. We loved sitting along the harbor with a glass of wine or ouzo at sunset. 
Day 8-9 -- ferry to Samos. This wasn't on your list, but I'd recommend it, because from Samos you can take a day trip to Ephesus, one of Turkey's most incredible archeological sites. Stay overnight in Samos somewhere near the harbor, and book a day trip for the next day. Highly worthwhile. Much better than a day trip to Bodrum or Marmaris from Rhodes, in my opinion. Pay attention when looking at ferries & Ephesus trips & hotels -- there are two main harbors. You will possibly come from Patmos to Pythagoriean, but your day trip to Ephesus might leave from Vathy (or Samos). And your ferry to Paros or Naxos will likely leave from Vathy as well. It's about a 30-minute taxi ride between the two harbors.
Day 10-11-12 -- ferry to Paros or Naxos or possibly even Mykonos. You'll need to check ferry schedules, probably early 2017, to see which island has connections from Samos.  From any of these, you can take  day trip to Delos. I would probably stay on Paros or Naxos, if I had a choice, rather than Mykonos. 
Day 13, 14, 15, 16 -- ferry back to Athens. Get car for visit to Delphi and Mycenae. Come back to Athens for your last 2 nights. 
Travel light; a carry-on size suitcase is fine. Remember light jackets or fleeces for evenings. Aegean Air has tiered pricing, with some cheap tickets, then mid-range, then high. As soon as you have your flight into Athens, book the Rhodes flight, to get the best price. Book Athens & Rhodes hotels maybe 6 months in advance. The others you can reserve just a few weeks in advance, or even on the go. Don't send deposits; there's always a chance weather will delay a ferry and you'll have to juggle your schedule (maybe even change islands, which is how we serendipitously got to Leros!).