The drivers are crazy in Sicily!
SE: Cavagrande. Huge gorge that you hike down into, beautiful cold natural pools at bottom. Pure heaven and a must do if going in the summer.
Caseificio Borderi in Siracusa market - best sando of my life: the queue can be an hour, and generally I don't queue, but it's worth it.
Duomo (Piazza del Duomo) the church has a Baroque facade stuck onto the original Greek temple.
Fonte Aretusa (Via Picherali) a freshwater spring at the edge of the sea
Parco Archeologico della Neapolis -the Teatro Greco particularly
Out of town a bit, plan a day (bring a picnic) at Pantalica, a necropolis of over 5000 tombs dug out of the surrounding cliffs. Make your way to the bottom and follow the lovely cold stream until you find your perfect picnic spot (you'll come across some places deep enough for swimming)
Trattoria Casalinga: get the vongole
Book a tour of Palazzo Biscari. Principo Ruggero Moncada, the heir and tour guide, takes you through this gem of Italian ostentation with pride.
and, of course, Etna.
Piazza Armerina The Villa Romana– serious mosaic vibes
Ragusa Ibla a lovely Baroque town on its own little mountain and great gelato