Bottomline: You can visit Sicily in any period of the year, depending what kind of tourism you like best.
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HIGH SEASON
In July and August the beaches and the streets teem with tourists and locals, for whom there are several entertainment opportunities: beach establishments (known here as lidi) offer various water and beach activities, one can dance at nightclubs on a daily (nightly) basis, water and amusement parks open their doors, many local festivals and religious processions are held. Among these, the Palio dei Normanni in Piazza Armerina is particularly famous. It is held August 12-14. For a change, or if the heat becomes excessive, you can hike on Mt Etna.
MID SEASON
If you visit Sicily in April-June or September-November, you will find mostly pleasant weather and uncrowded beaches. You will be able to visit the main sightseeing locations at your pace. You will also find numerous sagre, which are festivals generally centered on a specific product or event, such as the pistacchio in Bronte, the funghi porcini in Fornazzo, the wines in Milo, or the grape harvest in Piedimonte Etneo. The Ottobrata zafferanese is held in October in Zafferana Etnea, offering different festivals dedicated to local products, arts, and crafts.
LOW SEASON
In December-March, the weather is of course colder, but not harsh. On mt Etna, skiing and other winter activities are possible at Rifugio Sapienza or Piano Provenzana, generally from January to March, but of course it depends on the weather. You may also experience the colourful Carnivals of Acireale (30m drive), Misterbianco (1h drive), or Sciacca (3h drive). In Catania, the procession of Sant'Agata, a mix of pagan and Catholic traditions, is held at the beginning of February. A popular event is the Festa del Mandorlo in Fiore (Festival of the Blossoming Almond Tree) which is held in Agrigento around the beginning of March.
Mt Etna
Rifugio Sapienza is a popular destination. From there you can get to the top of the volcano (only with licensed tours) or to some of the several nearby hiking trails and sightseeing spots (Schiena dell'Asino, Valle del Bove, Mt Zoccolaro). You can plan a picnic in the pine forest (Pineta Ragabo, Pineta di Adrano, or Pineta Monti Rossi) along the slopes of Mt Etna, where you will find barbecuing facilities. You can also visit the Alcantara gorges, and bath in the chilly and crystalline waters of the river flowing in the middle of ancient eruptive rocks. Then, you can make a tour of the nearby wineries in Castiglione, and its hamlets Passopisciaro and Solicchiata, or in Linguaglossa and Randazzo. In winter, you can go skiing or do other winter activities in Piano Provenzana (Etna Nord) or near the Rifugio Sapienza (Etna Sud).
Taormina
Taormina is one of the most popular tourist points in Sicily. It sits on a hill, some 200m above sea level. You can walk and shop around during the day. After dark, you can enjoy the nightlife or perhaps a concert in the Greek-Roman Theatre. If you prefer the beach, Isola Bella and the beach of Mazzarò are at the feet of Taormina, from where you can go by cable car.
Acireale
Acireale has beautiful baroque churches and palaces. It offers also the most famous and beautiful Carnival in Sicily, being also considered the third most beautiful Carnival in Italy. Besides people crowding the streets masquerading, its most famous characteristic are the allegoric floats and the flowered floats, which parade in the streets on the Mardi Gras. If your stay is in the Carnival period, you must not miss it.
Catania
The main town near Fondachello is Catania. There you have plenty of places to visit: the Cathedral, Piazza dell'Università, Monastero dei Benedettini, Castello Ursino, and many others. Nearby there is Acicastello, with a Norman castle, and Acitrezza, with its faraglioni, which the legend says were the stones thrown by Polyphemus against Ulysses' ships.
Other places
At 1h30m drive from Fondachello, you can find Siracusa, Lentini, Palazzolo Acreide, and the Necropolis of Pantalica. A little further in the interior, there is Piazza Armerina famous for its Roman villa and its mosaics, and Caltagirone, worth a visit for the beautiful ceramics. Further away, you may visit Noto, Modica, and Ragusa. Agrigento is famous for the remains of Greek temples.