Sicily - Where to spend 3 "bonus" nights?

I feel like spending six of your nine nights in Taormina and Catania is too many. I also don't think you need to stay in both cities. They're roughly 1 to 1-1/4 hours apart by bus and train. I'd choose one of them and consolidate my stays. Taormina is beautiful but painfully touristy when a big cruise ship is in port; I hope you'll be lucky on that count. I can't imagine spending 2-1/2 days there; I was ready to leave after 2 or 3 hours. But I know some people love it.

Catania has a nice historic center but is otherwise more rough-and-ready. There's a WWII museum I missed, and there's an interesting food market. I don't think it has a lot of top sights (I'm sure the interior of the duomo is worthwhile), but my trip predated publication of Rick's guidebook, and I haven't checked to see what he recommends in Catania. Overall, it definitely doesn't have as much charm as Ortygia. Some people day-trip to Etna from Catania. There's also the town of Acireale on the coast between Catania and Taormina; it's closer to Catania. Someone on the forum made a positive comment about it recently, and a Sicilian recommended it to me as picturesque and not full of foreign tourists. Alas, I didn't have time to check it out.

I believe you can also pull off the Villa Romana del Casale as a day-trip from Catania. It worked in 2015, but I realize transportation may not be running as frequently now. There's bus service from Catania to Piazza Armerina, the nearest town of size. (And picturesque, so worth some exploring.) In theory there was a bus from Piazza Armerina to the villa back in 2015, but I missed it and ended up taking a taxi. I believe that cost me roughly 20 euros each way. It's definitely not walkable--too far, and the road had no shoulders. With six people, even two taxis wouldn't be exorbitant. I spent enough time at the villa that I was right not to pay the taxi to wait for me. Aside from the magnificent mosaics at the villa, it's nice to see a bit of central Sicily on the way. If you opted to make a full day of it by car, you could include one or both of the seldom-visited central-Sicily towns of Caltanissetta and Enna. There are probably lots of other picturesque options closer to the direct route between Catania and Piazza Armerina, though.

I would spend more time in Siracusa than your tentative plan of three nights to allow day-trips to Ragusa, Noto, Modica and Scicli. (Some can be combined on a single day, even via public transportation.) Or you could perhaps spend a couple of nights in one of them (Ragusa would be my choice). There is train and/or bus service to all those towns. Not fast and not necessarily very frequent, but you could make it work. Or, since you are six, I think it would definitely be worth exploring arranging a private tour of some of those Baroque towns. You'd spend less time waiting for buses or trains and getting from the stations to the historic centers, leaving you more time to actually see the towns. In 2015 there were half-day (possibly also full-day) bus tours to one or more of those towns, but with your group I think a couple of taxis (or one large van-taxi) would be no more expensive.

Trains in Sicily are not fast. Be very, very cautious about planning any bus trip on a Sunday or holiday. There is very little service then. I found the buses reliable, but if you're boarding at an intermediate stop you need to allow for the possibility the bus will arrive early or late.

If this were my trip I'd probably plan something like four nights in Palermo and four nights in Siracusa with the final night somewhere in Catania, Taormina or some point in between, depending on when I needed to be at the Catania airport for my outbound flight. Palermo has a lot of sights--the incredible cathedral in Monreale (via city bus), the island's best archeological museum, two enticing food markets and many churches and chapels. A day-trip to Cefalu is easy by train.

acraven thank you for your comments. We intend to plan day trips from Catania to Agrigento and another day trip from Catania to Etna. The "kids" will be coming in from the States to Catania and it will be a really long travel day for all of them (on the same flights). So, the first day in Ortigia is really just a walk about and a sleep.

It looked to us as though the quickest way to both of those day trips would be from Catania. I did want to see Palermo, but the flights / ferry from Malta was not helpful.