What you must do during summer is to dive or snorkel at the Faraglioni. The water is crystal clear and swim in such water is something very special. But you must be careful with the currents.
During autumn, winter and especially during spring, trekking on Lipari gives unforgettable experiences. The Faraglioni photo was taken during such a spring day walk.
Scooter and vehicle rentals are available for hire near the waterfront. A drive around the island can be quite pleasant - visiting the pumice quarry, other towns, beaches and lookout points.