We are 2 couples from the Midwest U.S. trying to put together a last minute visit from 8/28 to 9/12. We haven't booked air yet. Most likely we will fly in/out of Rome and use local airlines/trains/drivers/rental car to move around. Places we want to see are Sorento, Pompeii, Amalfi Coast in the Naples area then head to Sicily where we have no idea of an itinerary yet. We aren't looking for a tour but will put something together ourselves. Any suggestions as to best (not necessarily cheapest) transportation options from Rome to Naples and from Naples to Sicily and Sicily to Rome? Also, is Amalfi coast something you could cover as a day trip while staying in Sorento or best to spend a night or two? Any other suggestions or thoughts would be great.
Sicily merits a minimum of two weeks at least. Or do you only want to see only one place there? If only one, I recommend historic Ortigia Island , part of Siracusa.
The Amalfi Coast, Sorrento, Naples, Pompeii, Capri will take a good amount of your time. I added a few places in the area that usually are included by travelers.
First, are you happy with a quick "drive by" type itinerary? If not, follow Suki's suggestion, and just see Sicily. You won't run out of things to do in your time there, and it takes time to get from the Midwest to Sicily (the connections aren't the greatest).
If you do want to combine the Sorrento/Amalfi Coast area with Sicily, in the short time you have, don't fly in and out of Rome. Fly into Naples and out of Sicily, or vice versa (Naples to the US is a bit easier than Sicily to the US for flight options). If you don't want to see Rome, don't fly there, unless you need an overnight there to make your connection home work (I did, but I was using miles and had few options).
Note that in Sicily, while Palermo is the bigger city, Catania is the bigger airport, with more flight options. Between Naples and Sicily, you'll want to fly, although there is an overnight boat between Naples and Palermo. There is also an overnight train; one poster here liked it, another is quite vehement that it was miserable. I'd fly; there are multiple flights a day on these routes. To find flights, use Skyscanner: https://www.skyscanner.com/
Some consider a day trip from Sorrento to the Amalfi Coast highly worthwhile. I consider it a waste of time. The towns on the Amalfi Coast are small and very busy, and won't be fun if you're just whizzing through (at least, they wouldn't be for me).
If you want a VERY detailed report of how I spent 10 nights in Sicily, running around with a car and still seeing only a small part of the island (it was a great trip), here's my trip report from 2014: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina
If you opt for Sicily, fly in/out of Sicily. Sure, you'll have to change planes somewhere, maybe even Rome, but it's much better to fly on one ticket in case of delays/missed connections. I'd go with into Catania and out of Palermo for two reasons. First, it's best to plan a one-way route, rather than backtrack. Second, Catania's airport is forced to shut down unexpectedly when nearby Mt. Etna erupts. If it's closed when you fly in, there are alternatives. If it's closed when you fly out, not so much.
I'd lock down an itinerary and accommodations before booking flights. Sorrento and the Amalfi Coast are very popular and it may be next to impossible to find rooms for 2 couples in the same place that fit your budget. If you go for the AC, check before committing. While you can get to/from any town/village by road, the roads are often very congested, driving is miserable and parking even worse (both expensive and hard to find) and buses are packed with long lines of passengers left waiting for the next bus. You really need to choose a place that is a short walk from a ferry pier - and not all the places on the coast have them.
It's extremely late to plan this trip, at a peak time of year. You may wish to move it entirely to September, but Sorrento will still be heavily booked. As noted, Sicily, while not an "underdeveloped country", requires additional care and planning for a smooth visit. It is important to understand how hot this area is in August, and how much less common air conditioning is in Europe, and even, southern Italy. We were glad to have A/C in Sorrento in late May, and happy to pay for a more expensive hotel.
For the sake of speed, you may wish to use this Forum's search box, because visits to the Amalfi Coast from Sorrento, getting to Sorrento, and getting to Sicily are frequently asked about here. You can filter your results for One or Two Years old, to be sure of the freshest data. The time (and "Forum Results") filter appears in your search results, not in the initial Search.
We are now pushing back to late Sept thru Mid Oct, perhaps 17-18 days. That should give more time to plan, more days available on the trip and a little less crowded. I can already see lower airfare options. Thanks for the advice.