Advice on flights from Rome/Florence to Sicily

Rome (FCO) has more choices to/from Sicily with several flights with Alitalia, Vueling, Ryanair.
Florence (FLR) is connected to both CTA and PMO with Vueling with one flight a day, but not every day of the week.
Pisa (PSA), located 50 miles from Florence, has flights to Sicily with Ryanair with a daily pair to CTA, PMO, and also TPS.
Check the websites of the above mentioned airlines for scheduled flights and pricing.

Buy when you find a price you can live with. Then don't look back.

In my experience, Alitalia has comfortable seats and a better baggage allowance than the budget airlines.

Since FCO Roma is not far from Firenze, and there are two direct trains a day from the airport, I think it might be less stressful to take the train to Firenze on the 5th, then fly to Sicilia, work your way to the opposite coast and depart from there to spend your last nights in Roma, and fly home from FCO. A plane delay on your way to Italy can happen, so a relaxing train ride to Firenze can be a nice start to your trip, rather than the stress of catching another flight the same day.

I was thinking the same thing about trying to catch another flight to Sicily after arriving in Rome. I am leaning toward the train ride to Florence and then fly to Sicily. For what I am saving on the transatlantic price I can be a bit more flexible with the other flights.

Go to Sicily between Florence and Rome. Connecting to another flight (inbound or outbound) on a separate ticket is very risky. If you are delayed and miss your connection to Sicily, neither airline is liable to help you. Chances are you will lose the tickets and have to scrounge for places at top prices on the next flight to Sicily. If you are delayed going back to the US, the financial damage could be a lot worse.

If you are planning to go from one end of Sicily to the other, it makes sense to fly into one and out of the other, doesn't really matter where you start. First figure out how much time you have for Sicily and what you want to see there. You may find that you only have time for one side. Go to the wiki page for each Sicilian airport and scroll down, you'll find a table listing all the airlines that use the airport and where they fly to. Also consider Bologna's airport which is also close to Florence.

"Nickel and dime" - one way to look at it. Another is that you're ordering ala carte. Don't pay extra to choose seats for a one hour flight, does it really matter if you sit together? Maybe you only need to check one bag for the 3 of you. Just be sure to add it all up when comparing prices. I don't know about short flights on airlines like Alitalia, but on the budget airlines, prices only go up.

I like to fly out of Catania because of the view of Mount Etna from the passenger waiting area.

I agree with others.
Train to Florence upon landing. Visit Florence.
Flight from Florence to Sicily (Vueling).
Visit Sicily.
Flight from Sicily to Rome.
Visit Rome.
Return home.

If a flight from Florence is available on the day you need, don't bother with Pisa or Bologna. The extra time is probably not worth the airfare savings, if any.

For Alitalia domestic flights (not international ones), use the Alitalia "International" website. This has an English language option, but gives the same prices as the "Italian" website which is in Italian only. The US website shows much higher prices - for the exact same flights. Note that the cheapest tickets sell out fast (and some are probably already gone), so book as soon as you know your dates. Do compare prices on all airlines flying a route, but note that since Alitalia includes one checked bag and has a more generous carry-on allowance than the "budget" airlines, it can work out the same or cheaper. If you are using a budget airline, accept that you will probably have to check a bag. If you pay for the checked bag in advance online, it's much cheaper than paying for it at the airport.

I like Roberto's itinerary (Sicily in the middle, avoiding flight hassles). But if you do want to go to Sicily right after landing in Rome, you may want to do what I did. I booked one ticket from Rome to Palermo three hours after arrival, and one ticket ten hours after arrival. That way, I knew I'd use one ticket and throw away one ticket, and considered the cost of the trip to be the price of the two tickets together (which, since they were bought months in advance, was still substantially less than a last minute ticket on that route for my travel day). In the end, I made my three hour transfer with time to spare, but of course, I could just as easily have missed it. On the way home, I had to spend a night in Rome (since there was no way I was going to take a chance of missing the Italy to US flight using separate tickets). Again, Roberto's itinerary avoids these problems, and allows you to enjoy Sicily without worry.

I went to Sicily 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

I like the idea of going to Florence first, flying to Catania and hitting highlights there and then going to Palermo (splitting the week up between the 2) and then flying to Rome from Palermo. We will be in Rome before our flight home. There does seem to be more flights from Florence (and/or Pisa) to Catania then to Palermo. Agree that not worth paying for seat selection on the economy flights. Did anyone have any issues with checking bags in regards to size?

Sometimes they check weight, sometimes size, sometimes both, sometimes neither. So you either follow the rules scrupulously and they aren't checking and you could have saved the money, or you take a chance and they check you and you end up paying more. Sometimes the big airlines check, sometimes the budget airlines don't check. You won't know until you're at the airport.