Panarea is reached by boats from Lipari, largest of the Aeolians. The SIREMAR line  offers fast hydrofoils, which are preferable to slower ferries, particularly since you'll have no need for a car.
It's a small island, and you can get around quite well by foot.
Panarea has excellent, scenic beaches. Some offer views across the sea to the volcano-island of Stromboli; you can see ash and steam clouds from the volcano's frequent, small eruptions.
If a hiker, check out the Bronze Age ruins a short-to-medium (5-km round trip) walk from the dock, at the entrance to a nature reserve. No amenities, and the ruins are largely unexcavated; you'll be reminded more of the austere sites of the American Southwest than of the glories of Rome.