Catania is built of the thing that keeps trying to destroy it. Mount Etna's lava reached the city walls in 1669 and filled the harbor, and in January 1693 an earthquake killed almost two-thirds of the population. Catania rebuilt itself in grand baroque style, in black lava stone and pale limestone. That rebuilding is why parts of the center, like the rest of the Val di Noto, are on the UNESCO list. It's loud, dark and a little rough around the edges, and nothing like the polished Sicily of postcards.
This guide has three day trips in five days, and that is deliberate. Catania is a regional base: Etna, Taormina and Syracuse are the three big sights of eastern Sicily, and all of them are an hour or two from here, so a city-only plan would leave the best of the region unseen. The city itself needs two days: a fish market in a tunnel under the old walls, a baroque cathedral square, one of the largest Benedictine monasteries in Europe and a main street that points straight at the volcano. If you'd rather spend more time in town, drop any two of the three trips and give those days to the city; Taormina and Etna are the easiest to lose.
The plan is two days in town, then one trip a day: Etna on day three, Taormina on day four, Syracuse on day five. Start on a Monday if you can. The fish market has no official hours and is most likely closed on Sundays, so a Monday start puts it on your first morning, and it keeps Ortigia's market for a Friday. If you start on another day, check the market first and swap days one and two where needed. Keep day three flexible, because Etna depends on cloud, wind and what the volcano is doing, and swap it with day two if the forecast is poor. Western Sicily is a different trip: it has its own plan in 5 Days in Palermo. For the country as a whole, see Italy.