Sicily

I used this website and it has tons of info about all of Sicily, travel, lodging, sites, food etc.
Best of Sicily

You have a limited amount of time, so prioritize your interests and be aware of your travel style: driving or public transportation, changing cities every two days to cover a lot of ground, do you need a beach, do you like history, art, etc?

You could spend three nights in Palermo, day trip to Erice, Monreale, Agrigento or Segesta, stop in Cefalu on your way to Taormina, two nights there, last night in Catania. Leaves one night for somewhere else (Siracusa?), or add it to Taormina.

I would recommend getting a Lonely Planet book which will give you a good sense of potential itineraries. You should prioritize and stay focused on one area because 7 days is not much time.

I'd consider renting a car for your time outside Palermo if you can deal with a manual transmission; just verify that there won't be a significant drop charge for a one-way rental. There is much to see in Sicily, and you don't have much time; a car would allow you to move around more efficiently. The trains aren't especially fast, and there are few inter-city buses running on Sundays.

Do research the sights and build a list of priorities. I had at least 14 days for my trip last year, and I never got to the SW part of the island. Ancient ruins aren't my thing, so I skipped Agrigento and Segesta; did see the Greek theatre in Siracusa while I spent four nights in Ortygia, the old section of the city.

Your basic decision may turn out to be how much time to spend in Palermo. It's a big, bustling city just full of incredible sights, many of them religious buildings. Four nights are easily justified, but that would really curtail the rest of your trip.

Seven nights is a very short time in Sicily, so you will have to be brutally selective. Good for you for flying into Palermo and out of Catania.

My VERY detailed trip report of 11 nights in Sicily in April-May 2014, with both travel ideas and logistics tips, is here: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

BTW, you certainly can rent an automatic if you want or need one..

Even if you're not taking a RS tour, check out his tour itinerary. It really does hit most of the must-see stuff.
You'll have a wonderful time! Sicily is beautiful, and SO affordable (and the food is yummy!).

I spent nine nights in Sicily in May, also flying into Palermo and out of Catania. Spent three nights in Palermo and two nights each in Ragusa, Siracusa and Catania. I got around by bus and train. Public transportation was not very reliable. Half day (or so) trips to Modica and Noto turned into almost full day trips due to no show buses and trains. That being said, everything else was great. Food, wine, weather, people and sights. I thunk Ortygia was my favorite place, Palermo was also a great time.

We are spending 9 nights this fall in Sicily landing in Palermo and flying out of Catania. I am renting from Hertz- an economy size car with manual for a week at the downtown Hertz in Palermo for 191 euros, which includes collision and theft. We are leaving from Palermo to Trapani then Agrigento to Syracuse then Catania before flying out to Naples.