Greek islands run the gamut, and which you should visit and for how long depends on what you want from your island stays.
Santorini is the epitome of touristy, and Fira and Oia, the best-known of the villages, are sleek and polished. When the cruise ships are in port, Fira and Oia become uncomfortably crowded. It's better outisde the 10-4 period when most of the ships are there. One great sight is the ruins of Akrotiri, which date back to Minoan times (~3000 years BC), and a great complement to that is the museum in Fira, which has some of the wall paintings and found items from Akrotiri. If you are looking for a "cultural" sight on Santorini, this is probably the best. As for the rest of the island, it is a tourist haven, and precious little in the way of typical Greek life takes place there.
Naxos is the opposite, an island without a deep water harbor that consequently is not visited by cruise ships and thus retains more of its authentic Greek flavor. There are ruins to see and good beaches, and charming villages in the interior as well as the main port town. If it's not too windy, you can get a day excursion to Mykonos and the archaeological island of Delos. A similar alternative to Naxos is nearby Paros, with its little-sister island Antiparos a quick ferry hop away. Either would be a good counterpoint to Santorini.
I agree with heading straight to Santorini upon arrival in Greece. If you just want to snap some lovely photos and see Akrotiri, two or three days will be adequate; if you are someone who can luxuriate in gorgeous views for hours, you may want to spend longer. Ferry to Naxos or Paros; there is more to see and do on these islands, so I would spend more of your island time there than on Santorini. Two or three days for Athens will round out your trip.
May is wonderful in Greece, and other than the most popular cruise ship targets, not crowded at all that month. I spent four weeks in Greece in May and had only a day and a half of rain the whole time; mostly the weather was sunny and the skies were blue. It was cold and windy on Mykonos, but quite pleasant everywhere else and even hot on the mainland and in Athens.