We would suggest a 9-day tour...

DAY 1 Arrival at Palermo airport and transfer to your downtown hotel. Palermo by night with English-speaking guide.
DAY 2 Full day guided tour of Monreale and Palermo highlights. Overnight in Palermo.
DAY 3 Visit the west of Sicily: full day guided tour to Segesta, Erice and Trapani (to see the salt pans). Back to the hotel in Palermo for the overnight.                                                                                                        DAY 4 Leave Palermo, longway transfer to Agrigento for a guided visit to the Valley of the Temples.  Overnight in Agrigento.                                      DAY 5 Tranfser to Piazza Armerina for a guided visit of the Villa del Casale. Resume the trip to Noto. Overnight in Noto.                              DAY 6 Half day guided tour of Noto. In the afternoon transfer to Taormina. Overnight in Taormina.                                                            DAY 7 Morning guided tour of Taormina. Afternoon at leisure. Overnight in Taormina.                                                                                              DAY 8 Morning excursion to Mt Etna. Rest of the day at leisure in Taormina. Overnight in Taormina.                                                            DAY 9 Departure transfer to Catania airport in time for your departure flight.