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. Ferry lines to the Aeolian Islands depart from Milazzo (the closest to the islands, but most difficult to reach), Messina, and Palermo in Sicily, as well as Reggio di Calabria and Naples on the mainland. For international travelers, the most direct route is to fly to Catania, Sicily, transfer to an express bus to Milazzo, from where the most ferries run. As an alternative to the bus, Europcar offers a special deal to rent a car one-way between Catania and Milazzo, but this must be arranged for in advance.
The island is so small that there are no roads, but only paths. There are no cars on the islands, but only motorbikes, 3-wheeled trucks, and golf carts. There is a golf cart "taxi service" that runs between the port and all points in the only town, but for those who can walk up a hill, all points in the town are accessible by foot.
At high season, it is necessary to book far in advance. Most accommodations on Panarea are open only from May through September. In My through mid-July, and in September, you will be met at the dock by representatives of various hotels and inns, and you can negotiate a price for a room with a view of the sea. Hotel Lisca Bianca is a good-priced hotel in the main port with a lovely veranda where breakfast as served. The proprietress can also offer rooms in villas nearby.