Since the distances between islands are substantial, island-hopping by boat can be costly and time-consuming. Add the fact that each island is quite large, and it becomes clear that you're best off choosing one or two islands to explore in depth rather than attempting a grand tour. Both Olympic Airlines and Aegean Airlines offer some interisland flights that are relatively inexpensive, frequent, and fast. If you travel by ferry, you'll find that departure times are more reasonable for travel from north to south, whereas traveling in the opposite direction usually involves departures in the middle of the night. Fly to Lesvos (Mitilini) if you insist on seeing all three. The islands are too large and the roads often too rough for mopeds to be a safe option; because the bus routes and schedules are highly restricting, you'll find that if you want to get around it's necessary to rent a car.
Museum & Site Hours Update -- If you visit these islands during the summer, check to see when sites and museums are open before setting out. The officially posted hours are not always maintained and most are closed 1 day a week.