where to go and what to do from Rhodes?

For a day trip/boat excursion from Rhodes, I would recommend the island of Symi. Be sure to have your cameras ready as you near the island ... Symi has one of the most beautiful harbors in Greece. You'll probably want 4-5 nights to see Rhodes. A car would be useful for one or two of those days. You can find lots of sandy beaches.

For another island besides Rhodes to stay on, I highly recommend Kastellorizo. This is a tiny, absolutely spectacular island. This link is to one of its lovely hotels. The best way to get there is by flying from Rhodes; check Aegean Air -- there may not be flights every day in May. Kastellorizo does not have a sandy beach, but you can swim (often with sea turtles) in the deep water harbor. You can take a boat to a blue cave just outside the harbor. You can take a day-trip excursion to Kas in Turkey. You can learn the history of the island in the free documentary that that is shown daily. We absolutely loved Kastellorizo. Totally relaxing, wonderful restaurants. I'd suggest 4-5 nights here too.

Then fly back to Athens for 2-3 nights.

The white buildings with Blue trim are primarily found in the Cyclades island chain which is not close to Rhodes. If you want to visit one of those islands you will have to fly from Rhode to Athens and take a connecting flight to the island of your choice. Santorini might be a good choice, it is spectacular. For a more authentic Greek experience look at Paros or Naxos.
We spent a week in Rhodes and had not problem keeping ourselves busy.
Lindos and Rhodes http://www.flickr.com/photos/stanbr54/sets/72157634754696058/
Trip around Naxos http://www.flickr.com/photos/stanbr54/sets/72157634605629689/