1. Catania: the guidebooks have a lot of info about the city. Not to be missed are the Via Etnea, via dei Crociferi, Monastero dei Benedettini, the Teatro Massimo, the Duomo, Piazza Universitá, Villa Bellini
Ognina is where you can have a promenade along the seaside.
Nightlife in Catania is very mediterranean, and it is known as Movida (in honour of our spanish governors (?)(!)). Young and university nightlife takes place around Piazza Teatro Massimo.
Catania enjoys great street food. Very typical is a sandwich with horse meat. There is a place very close to piazza Teatro Massimo where they make it, but any place is ok I guess. Try the very typical pasta alla norma and the arancino.
2. Taormina, Giardini Naxos, Castelmola: pure beauty. Amazing view of the coast from the Ancient Theatre. Prices are very expensive here, as the town is known as a VIP and cinema destination. Close to Taormina, up on the hill, there is a very small and typical sicilian village: Castelmola. Great view. Not to be missed. Bar Turrisi is the place in Castelmola, you'll find out why. Giardini Naxos was originally a greek colony. Nice promenade along the seaside.
3. The coast between Acireale and Catania: Acireale is nice town, with a nice church. Between Acireale and Catania there are a lot of small towns with very nice restaurants. For a very good pizza I recommend the restaurant 'Acqua del Ferro' (http://www.aquadelferro.it/). Not to be missed are AciTrezza and AciCastello (with its wonderful Norman Castle, witnessing the presence of North-Europeans in Sicily). Some of the most famous italian novels are based on characters from Acitrezza, due to the great novelist Verga. For fish you can try restaurants in Acitrezza or Capo Mulini, I don't have a particular suggestion.
4. Etna: this should probably go first. The largest volcano in Europe. There are two ways to go up: on the south side (above the town of Nicolosi, Rifugio Sapienza) or on the North-East side (above the town of Linguaglossa, Piano Provenzana). Cars can get only up to ca 2000 m (Sapienza). From there on, if you really want to climb up to the central crater, you need to go with a guide (e.g. http://www.guidetnanord.com/it/ from the North-East side). There are a lot of very nice and ancient town around the Etna volcano. I recommend Randazzo, Zafferana (try here the pizza siciliana in the two bars of the main square), Bronte (town of pistacchio). Near Linguaglossa there is a pizzeria where they serve huge pizzas (https://www.tripadvisor.com/Restaurant_Review-g660768-d2272201-Reviews-Pizzeria_Contrada_Sciaramanica-Linguaglossa_Province_of_Catania_Sicily.html).
5. Siracusa: absolute gem. Beauty, history and melancholy. Siracusa is the greatest witness of our Greek past. Guidebooks have so much about it that I do not need to say more. The greek theatre cannot be missed, as well as the Orecchio di Dioniso. I also suggest the Castle of Euryalos. There are a lot of nice restaurants in the centre.
6. Ragusa: probably the most typical sicilian town. Ragusa Ibla is the ancient part of the town, quite chic nowadays, but still not to be missed. Close to Ragusa, there is Marina di Ragusa, with its nice beaches. Around this area there are beautiful sandy beaches. Around Ragusa there are some beautiful villages: Scicli and Modica (try the typical chocolate at the Bonaiuto Café) are the most famous ones.
7. Noto: town of Sicilian baroque. In May there is the famous Infiorata, but I guess you won't be there on the third week of May.
8. Beaches: the East Coast has many rocky beaches, which the locals love but probably are not so nice for tourism. However, Fondachello (in the municipality of Mascali) and Letojanni have more sand-like beaches. In May there are not going to be many people and the water is probably still a bit cold, but I would try nevertheless.
9. Sicilian sweets, confectionery: Sicily has great cuisine, but that's common to every region in Italy. It is its mastery in sweets and confectionery that is unmatched. I think it originates from the blending of the Arab and French masteries. Try: cannoli (expecially with ricotta), zeppole di riso (my favourites), cassata (which is originally from Palermo, but we also do it good!) and many other things. In Catania a good place is Pasticceria Savia, in Via Etnea.
There are many more things to do, see and eat. I surely forgot some (maybe many!) The West Coast is a jewel, but I don't know much about it. The Valle dei Templi close to Agrigento is a must. Cefalú, Palermo, Messina are also musts. If you have time you should have a look at the Eolie Islands or at the Egadi Islands.
Finally, what makes Sicily so incredibly impressive is not just the sun which shines strongly without pause, not just the warm sea which welcomes the tired bodies of the people in summer and beats powerful on the rocky shores in winter, not just the red of the lava which lazily flows down the Etna, not just the time that goes on slower than in any other place.
Sicily is the symbol, the gem, the glory of the Mediterranean because of its blend of ethnies, cultures, histories. Every town tells the story of a different migration: the Greeks seeking new lands, the Romans learning from them and cultivating wheat, the Normans coming from far away to warm their skins, the Arabs knocking at the doors of Europe, the French looking for their mediterranean roots, the Spanish surprised to find such a similar land, but where everything is more extreme, the Subsaharians coming today to seek a new, peaceful future.
And finally the Sicilians, unconsciously carrying the weight, the languages, the smiles, the behaviours, the traditions of centuries through a new sunny day.