10 days in Sicily

Palermo on the west coast and Siracusa on the southeast coast. These are big cities, but have good bus and/or rail connections to places nearby.

If you can, I suggest flying into Palermo, four nights; Siracusa, four nights, and fly out of Catania (two nights so you can daytrip to Taormina on one of them). There are shuttles (Alibus) to/from the airport in Palermo and Catania.

I agree with Zoe.

You want opposite sides of the island, in places big enough for good transit connections, and places you want to visit in and of themselves. Palermo and Siracusa fit the bill perfectly.

Thanks for all the suggestions

I will recommend Catania and Palermo.

For the first (or 2nd) leg of your trip use Catania as your home base. From there you can hay day trips to Siracusa-Ortigia, Aci Castello, Aci-Trezza, Raguzza, Notto, and perhaps Agrigento to see Il Valle dei Templi, which in my opinion is a must. Best keep greek ruins. You can also visit Piazza Armeria to see Villa Romana del Casale. Taormina is super beautiful with the Roman-Greek theater, walk thru the little back streets and sip local wine.
Uphill Forza da' Agro' little town is near by where The Godfather wedding was film.
In Catania, the greak anfitheather right in downtown Catania, Il mercato de Pesce, stroll via AEtna, do the passegiata and eat awesome seafood. For drinks go the the "collegiata" area, lots of wine bars.

Palermo - go visit Segesta, San Vito lo Capo in Trapani, Erice, Il antico mercato in downtown Palermo. The Egani Island ferry is near by.

If time permit, visit the island of Malta. It is a short/cheap trip to north Africa but you will enjoy it.

Thanks for all the suggestions