Sicily

Personally, I would not stay in Taormina since it is away from some of the sites you might want to visit. I would stay in Ortigia. From there it’s easy and worthwhile to visit Ragusa, Modica, Scicli and Catania. Depending on how far afield you’re willing to travel, consider visiting Agrigento and the Roman Villa outside Piazza Armerina. On my first visit to Sicily, I stayed in Catania and spent the better part of a leisurely day driving up to one of the ski areas on Mt Etna and then circumnavigating the volcano stopping in a few towns along the way.

We didn't love Taormina. Very busy and more expensive than other cities. I feel like there are more interesting day trip options from Siracusa/Ortigia, which also has quite a variety of sites. My other suggestion is Palermo. We were surprised how much we really enjoyed Palermo. It is a large city and a bit "gritty" which makes it super cool to visit. We loved the markets and the churches. There are a variety of day trip options from Palermo. Food was interesting and fun in both Palermo and Ortigia.

What do you want to see in Sicily? Have you done any research? Your answer would be the determiner of whether you concentrate on the eastern or western side of the island. You are smart in realizing that with just one week it is best to concentrate on one general area and not rush from place to place trying to see all major sites. From Rome there are easy connections by air to either Catania or Palermo. From Sicily you say that you want to travel onward to another destination in Europe. Do both of these cities offer direct connections to where you want to go? If not, that will influence the decision. How do you plan to get around once in Sicily? There is so much to see and so many options that you need to know which sites you really prefer trying to see and then you can plan your itinerary. Transportation is also a big factor. Join day tours? Bus? Train? Rental car?

My personal first choice would be for Palermo and from there you can consider daytrips to the ruins at Segesta, medeval Erice, beautiful Monreale, pleasant Cèfalú, nature and seaside activities at Castellammarel Golfo, Scopello and the Zingaro Preserve. Even a daytrip or an overnight trip to Agrigento could work. In Palermo make sure you do a street food tour and visit the markets. You will have great food options all over the island.

If you choose the eastern side, I would consider two bases, Siracusa/Ortygia and also your Taormina. From Taormina you can visit Etna, the vineyards on the slopes and the Aci towns on the coast. For hiking and nature you can also consider the Alcantara Gorge. From Siracusa ( Ortygia is beautiful ) you also have all the nearby lovely Baroque towns.

Look at youtube videos and Sicily documentaries, guidebooks, and tons of great Sicily travel accounts on Instagram to get a better idea of what appeals to you and then make your plans.

Since you have limited time, I would choose one area and it would either be Palermo or Siracusa. After having lived in both, my preference would be Palermo. As others have said, it is bigger and grittier but I think it offers a wider variety of experiences. Siracusa is smaller and has great sites, too. For me, Palermo just has more. Siracusa might be more relaxing, though, if that’s what you are targeting.

We just got back from Sicily/Rome, we had 9 nights in Sicily and did Taormina, Siracusa and Palermo with 3 nights in each. I liked all 3 for different reasons. We very much liked Taormina, it was relaxing, not crowded in April, and not more expensive in our opinion. We stayed on the island of Ortigia in Siracusa and it is charming to walk the small streets etc. we went to the Greek theatre there too.
Palermo does have more sights. We felt we had left some things unseen in Palermo with 3 nights, where as not so much in the other two. As other posters have said, it depends a bit on how you like to travel. For a more relaxing comfortable holiday I would split my time in Taormina and Siracusa, using Catania's airport.

I’d vote against Taormina for a number of reasons, but also because you’ll be there in early spring. There’s no guarantee you’ll have great weather, and it’s a resort town. Siracusa (Ortigia) is a hidden gem with some actual sights and is a better launching off point for day trips. I had four nights there and it was a delight.

And adding to the chorus, Palermo is awesome. Great city to explore. “Gritty” is not the word I’d use, Palermo has faded elegance. I found the city to be classy, sophisticated, hip and clean. Day trips to Cefalu and Segesta are easy options, Cefalu you could do yourself by train. Of course, Monreale is not to be missed. Some folks consider that a “day trip” as it’s technically out of town.

Another vote for Palermo. We were very surprised by how much we enjoyed the city. So much to see and do. We spent 5 nights there with day trips to Monreale and Cefalu. If pressed for time I would skip Cefalu.

Depending on your mode and speed of travel, based on our ten day self-drive trip, this may be an option:

Arrive Catania, drive to Taormina via Castelmola

Drive an Etna wineries loop (Emilio Sciacca), back to Taormina

To Ortigia

To Agrigento via Villa Romana del Casale

Agrigento

To Palermo via Monreale

Palermo

Unlike others, we liked Taormina…not crowded or pricey. The Valley of the Temples at dusk and Monreale were highlights for us. And, Palermo was delightful and delicious.