Travel is a mind-broadening activity. That is why many of us do it. Sometimes the experience is wondrous and overwhelmingly positive; sometimes the learning can involve more mixed experiences. Sicily will give a mix.
The island is such a parodox, and it is difficult to work out, at first, why it is the way it is. For starters, it has more going for it as a travel destination than most places. A great central Mediterranean location makes it accessible and gives it a nice long season. It’s well-served these days by cheap flights in and out (with Ryanair). Its beaches are wonderful, it is home to the largest active volcano in Europe, Mount Etna, it has some lovely little islands offshore, great diving locations on the north coast, and it possesses a depth of historical and archeological sites from a range of diverse civilizations that has no real parallel in Europe.
So you wonder why it does not wipe the floor with Cyprus, Ibiza, Corfu, and others as the island destination of choice? Why does it make so little effort to exploit its potential? Major sites are served by a pitifully inadequate public transport infrastructure. Parking is largely a nightmare. Attractions are rarely supported by the sort of side-show that encourages you to stay a bit longer. Mt. Etna, for example, the largest active volcano in Europe, has one crappy souvenir shop and a small bar/restaurant at its base camp, Etna-Sud. No museum, no additional information, no marked paths, nothing. So you ask yourself, "What's wrong with these people?" The island is not wealthy and there's a lot of unemployment, why does it not take the easy step of beefing up its attractions and make some relatively easy, and much needed, money? The sad and simple reason was explained to me by a Sicilian. Who'd be an entrepreneur when half your profits are siphoned off by your mafia "sponsor"?
Basically that means, with the exception of straight beach holidays on the east coast, there is a bit of work to be done if you want to get the most out of your trip. Go east, west, or south and you need a car, a good guide book, and detailed map to find the gold. Signs are generally poor, so don't be surprised to find what you are looking for before you see a brown sign leading you to it, and don't be surprised either if there is nowhere to park and no toilets. Think ahead and stock up on water and snacks, especially on a Saturday night. Even in Palermo, you'll find that most places, even restaurants, don't open on a Sunday.
Think carefully before you plan to spend too long in the bigger towns of Palermo, Catania, and Messina. There might well be quite a bit to see, in Palermo especially, but you may find that the bustling, gritty atmosphere and the incessant noise will grind you down. Whilst Palermo is busy and densely populated with plenty to see, do, eat, and drink, "vibrant" is not an adjective that fits it very well. We spent four days there and wished for home after two.
Relaxing times can be had in the east - of that there is no doubt. Taormina and Giardini Naxos do allow a person to chill. These are the most obvious locations that encourage a stay of more than a day or two. Although the process of sightseeing can be stressful and problematic, the sights themselves are generally worth the effort
Money, Payment, and Banking Issues are things that must be carefully thought through. Travellers Cheques are not recommended because they are a nightmare to change. Credit Cards are not as widely accepted as they are in most other western European destinations, with some fairly large restaurants and guest houses refusing to take them. Banks, even the really big ones near the Teatro Massimo in Palermo, are a nightmare. Queuing etiquette is largely ignored by customers, even if there is a formal ticketing system, with pushing-in and barging past common-place. Procedures at the counter are painfully slow and practically prehistoric. To give an example, I needed to draw some cash on my card in Palermo. When I finally reached the counter, the teller had difficulty swiping my card and getting an imprint on the paper counter-foil. This was fairly obviously because he put the card in upside down and was trying to get an imprint from the smooth side. He had six failed attempts in total (each time with me desperately trying to catch his eye, each time with him deliberately failing to make eye contact in a ridiculously exaggerated fashion) before he finally figured it out for himself. Then he had a problem because my signature on the form did not exactly match the one on my eight-year-old passport. After four failed attempts to recreate my old signature (which incidentally did not match the one on the back of my current credit card), he decided to hand over the cash, shaking his head gravely as he did it. My advice? Make sure you know your PIN numbers to avoid all of this nonsense.
Personal Security, I have to say, was less of an issue than I'd anticipated. We are not ones for wandering around in the middle of the night so we don't test it to its limits, but, provided you stick to wide, well-lit streets with people in them and the odd car going past, and you take the obvious sensible precautions, you won’t have any problems. I did not notice the fabled armies of roving pick-pockets in or around the airport or stations, and, what beggars there are in Palermo, are fairly passive. There are some pretty big stray dogs though ...
So, the conclusion. Take in the sights early in the trip, moving on after a day or two in each area (Erice, Palermo, Agrigento, Syracuse) - that will be enough. Leave some time at the end of your trip to chill out in Giardini Naxos, Taormina, or Cefalu. That's my advice. Do things in reverse and you may find that you go home truly worn and stressed out.