There are many ways of exploring Sicily, from visiting the bigger cities and tourist centres, food and winery tours or seeing the major historical sites from Greek temples to endless museums.
Logistically moving around Sicily is difficult simply because of the mountainous landscape, bad infrastructure, lack of reliable public transport and really confusing or absent signage. Rather than attempting to see the entire island in one weekend (which I assure you is impossible), the best thing to do is simply break the island into smaller pieces and explore a smaller part of it.
It is easy to hire a car from any major airport and together with a reliable GPS, a guidebook, a little research and some Italian, you can easily negotiate yourself around a particular area.
One trip or vacation to a concentrated part of the island is a perfect way to soak up the culture and colours associated with each of the nine different provinces (Palermo, Catania, Messina, Siracusa, Ragusa, Enna, Caltanissetta, Agrigento and Trapani.)
Western Sicily for example includes Trapani, Marsala and basically everything west of Palermo from Castellammare del Golfo around to the Aegadian islands, down the coast to Mazara del Vallo, if you want to be particularly challenged you can make it down as far as Agrigento (but I think Agriento deserves more time to be savoured and is best to be grouped together with central Sicily).
Sicily can be sliced down the middle from Palermo into its heart to Piazza Armerina, Enna, Caltanissetta down to Agrigento which is filled with much history, archaeological sites and festivities during the year.
Then there is North Eastern Sicily which can be done by car from Palermo along the coast towards Messina and can include visits to places like Cefalù, the Aeolian Islands, many small coastal and mountain towns around to Messina and the resort town of Taormina.
If you decide to arrive at Catania airport you can start from there and explore along the coastline as there are many fascinating fishing villages and resorts all the way down to Siracusa and Ragusa.
A few days to explore the Val di Noto towns inland from Catania will give you the chance to experience the eight Baroque treasures of south-eastern Sicily: Caltagirone, Militello Val di Catania, Catania, Modica, Noto, Palazzolo, Ragusa and Scicli, were all rebuilt after the 1693 earthquake and are filled with ostentatious architecture, breathtaking scenery and equally rich culinary landscape to taste.
From Catania it is easy to catch the Circumetnea a historic railway which takes you leisurely around the base of the Mount Etna around to the picturesque seaside town of Riposto. From Catania airport it is simple to explore Etna itself and the endless small towns near and around the Mount Etna regional park, this area also boasts world-class wineries, restaurants, historical sites endless farm stay or luxury bed and breakfasts, spas and a golf course.
Sicily is a multifaceted place with endless things to explore, simply do some research into whatever you may be interested in and see if you can explore the island through your hobbies and passions.
There is something for everyone Sicily is a paradise for people interested in hiking, mountain biking, nature photography, snorkelling/diving and windsurfing.
Sicily boasts some of the best beaches in the Mediterranean, if your family is originally from Sicily you can visit the town of your origins, foodies will have endless things to taste with a succession of Sagra food festivals throughout the year and the island has some of the best wine in the world.
There are literary parks to explore Sicily through its greatest artists, if you are after a luxury holiday there are many five star hotels and resorts, you can take a helicopter ride around the island, sail around the coast and hop around the surrounding islands, take archaeological tours around the most well preserved Greek temples outside of Magna Grecia, immerse yourself in the thousands of museums, palaces, castles, markets, religious or food festivals, squares, do an inspector Montalbano, Mafia or Caravaggio inspired tour.
The possibilities are endless simply break off a piece of Sicily and have a taste.