sicily & aeolian islands

Never been to the islands. Depending on how much time you'll be in Sicily you could visit quite a bit. I would suggest doing it by car if you'll be there any significant amount of time.
Taormina is quite nice to visit. I suggest taking a couple of hours to visit Castlemola which sits above Taormina. On the eastern half of the island you also have Catania, Syracusa, Messina. We haven't been on Etna but did drive to a few of the small towns surrounding Etna. If you have time Agrigento, Segesta, Palermo, Marsala, Trapani, San Vito lo Capo are just a few places that are nice to visit.

Also consider the ancient ruins at Selinunte. Have been to Panarea and Stromboli in Aeolians. Send me a PM for info.

I have stayed on both Stromboli and Lipari. I would recommend staying on Lipari and taking a late afternoon excursion boat to best see the Strombolian effect of the volcano from the water. The area around Zaffarena Etna is very handy for visiting the Etna volcano. It's also a pretty fruit- and wine-growing area. I loved going up Etna it's impressive. You can arrive on Lipari from Naples on an overnight ferry, then continue by a different ferry to either Palermo or Messina. Getting there from Naples also gives you the chance to see the Vesuvius. (Can you tell I like volcanoes?) A couple of Sicilian must-dos that may have been already mentioned (in no particular order): - the Greek temples complex in Agrigento - the Roman villa & its mosaics in Piazza Armerina - the hilltop town of Erice (near Trapani) - at least an overnight or two in Taormina - staying in an agriturismo in Scopello (Tenute Plaia) and swimming at an old tonnara - visiting a couple of wineries (i.e. Donnafugata) in Marsala - going up (not down) the ceramic steps in Caltagirone - eating the best pasta "frutta di mare" at Da Alfredo's near San Vito lo Capo Sicily is not like the rest of Italy, it really is quite distinctive and, in my case, addictive! I have found a car to be a real asset in Sicily and the driving there to be easy (except for Palermo).
I hope this helps! PS - pick up any of Andrea Camilleri's Inspector Montalbano books. No one can give you the "taste" of Sicily quite like he can, IMO! ;-)

Stromboli: the guided hike to the top of the volcano and the view into the erupting caldera is an unforgettable experience.

The islands are worth visiting but not before June and not after September. Etna is always worth a visit, along with Siracusa, Noto, Agrigento, Selinunte, Erice, Segesta, and Palermo. Messina is not worth your time. The Villa Romana del Casale is closed for renovations.