Need help with Sicily itinerary

I feel like I'm always making negative comments about Catania. I like Catania. I spent several nights there in 2015 (and none in Taormina). But, really, I don't think there's much comparison between Catania and Siracusa. For me, it's Siracusa all the way.

It doesn't sound like you're going to be hiking up Mt. Etna, so I urge you to do a bit of reading and decide whether you really want to spend a lot of time taking a tour to it, or driving around it, when you have so little time on Sicily. If the weather cooperates you can see Etna from far away at the cost of very little of your vacation time.

Train service along the Taormina-Catania-Siracusa route is perfectly usable, though you'll need to switch to one of the frequent buses up to Taormina from the RR station at Giardini-Naxos. Where a car would be useful on the eastern side of the island is for traveling from Siracusa to some of the lovely inland villages like Ragusa, Noto and Modica. Ragusa is especially interesting.

Although the (limited) train service in Sicily is mostly along the coast, there's a line that cuts across the interior from Catania to Palermo. But trains on that track are infrequent and take about 5 hours for the entire run from Siracusa to Palermo. You might or might not feel that it's worth driving that leg to avoid the train and make the trip in probably about 3-1/2 hours rather than 5. The Sicilian roads seemed OK to me from my seat on a bus, but you don't want to drive in Palermo (think Boston + New York City + Naples), and rural roads through small towns can be partially blocked by cars parked higgledy-piggledy.

There's also limited-frequency bus service between Siracusa and Palermo, which I think would be the easiest mode of transportation to use. The schedule shows that the trip takes 3 hr. 20 min. and that there are mid-June departures at 8 AM and 2 PM on weekdays.

To me the top sight in Palermo is the cathedral at Monreale, which is actually outside of town. But the city's markets are interesting, there's a good archaeological museum, and you also have a great variety of stunning religious buildings. Palermo's big and boisterous. You'll need to be care to pick a safe place to cross the street.

Aside from Palermo itself, I especially liked Trapani and especially Erice. But there are important Greek sites, too. They are not especially my thing, and I had already seen (and enjoyed) the Archaeological park and museum in Siracusa.

The former fishing village turned beach town of Cefalu is east of Palermo and easily visited by train.

I would add a few things to that very well detailed answer above.
In terms of greek sites, Segesta is wonderful and easy to do(thinking of your grand father). One great temple, take the little bus that brings you on top to see the roman amphitheater. It's a wonderful site can be done the same day as Erice, mentionned above, a great little medieval city.
The other major greek site is Agrigento, which take a good chunk of a day but is also impressive.
For the etna, you can do the easy tour. Car to cable car, big trucks that brings you up and then a rather easy walk with a guide. Expensive but worth it I'd say.
Do not miss Syracuse, especially Ortigy. Wonderful.
Do not miss Monreale as mentionned above, which is just outside Palermo. You could do it before staying in Parlermo.
In Palermo, do not miss the Palazzo dei normanni and the chapel, the food markets like the Ballaro market. It's a different planet.
Massimo theater is gorgeous
For something different, the catacombs dei cappuccini is a unique experience which we very much enjoyed, and which RS suggested in his tv show.
I always travel with rental car, so I can't about something else.
Aside from Palermo, driving was easy. Palermo is a different beast. :-)
You have to be calm, because people aren't really friendly drivers. It's a controlled chaos!

You've gotten some good advice, already: Taormina, Siracusa, Monreale, Agrigento, Cefalu, all super. Catania is not in the same class as the other cities mentioned.
I'll just add that having a car is very convenient, We've never been assessed a drop-off fee when returning a car in a different city in the same country. There are a few quirks to driving in Sicily (actually all of Italy). If you decide on a car, do a search on this site, or come back for specifics.
BTW, I'd cut a day from Florence and add it to Sicily.

Thanks for all of the advice. As far as Catania, I was thinking of that as a base because it is sort of in the middle of the East. From there we could do day trips north and south of it. There are also many more rentals available there as opposed to the other cities on the east. Please let me know if you know of specific rentals on VRBO, AirBNB or Booking.com or other that I should check for availability. I have something reserved through booking.com but I still have time to change if something better is available.

As far as the West we'll probably stay in Palermo or just out side of Palermo. Will avoid having a car in Palermo as there is very restricted parking, etc.

I wish I could subtract a day from either Rome or Florence but I already have airfare paid for. One of the "extra" days in Florence will probably be used to take a day trip to the Tuscany region to visit a winery.

I still have a lot of details to firm up so thank you for your help!

I was in Sicily with a friend in April-May of 2014; my VERY detailed trip report is here: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

There shouldn't be an extra charge for picking up a car in Catania and dropping it in Palermo. As you'll see from my report, we managed the driving fine.

If you can snag the place we stayed in Siracusa, it would be a good base for eastern Sicily. I loved staying in Taormina, but driving out of there is a bit harder (still doable). I didn't get to Catania, and Mt. Etna was fogged over on the day we were going to see it, so we just saw it from a distance (it's readily visible from Taormina or Catania).

For western Sicily, using Palermo as a base and dropping the car when you get there is a good idea. It should be too hard to drop a car in Palermo itself, since the car rentals are at the train station or in a newer section of town, and you don't have to drive through the old part of town to get to them (you definitely do not want to drive in the older sections). If you're hesitant, you can always drop the car at the airport, although that is a detour from the city. We didn't get to Cefalù, which everyone seems to love, so that could be an alternate base if you prefer a smaller place. Monreale was definitely a highlight.

I think Palermo, for all that it is hectic, is a better base on a short trip than Cefalu. There's far more to see in Palermo. If staying in Cefalu, you'd probably end up making at least two day-trips into Palermo. Aside from Cefalu itself, I'm not sure there's anything significant that is as easily visited from Cefalu as from Palermo other than Piazza Armerina in the interior, which Google Maps estimates is 3 minutes closer to Cefalu than to Palermo.