I'm usually not so dogmatic, but I can say that with such a short trip, and with the rest of your itinerary, a day in Positano is a complete waste. I saw Positano as a daytrip from Naples, and consider it one of my great travel "mistakes." It takes quite a bit of time to get to Positano from anywhere else, and it's not the kind of place you can enjoy in a short visit; the point is settling in, not blitzing through. Coming from Naples I felt it was a waste of a day; coming all the way from Rome, I would have been even more unhappy.
Then there's the fact that you're going to the Cinque Terre, so you'll get your seaside time. If you really want to see Positano, cut something else, and spend several days in the area (Sorrento, Amalfi, Capri, Naples - there's no shortage of things to see there).
In addition to flying into Venice and out of Rome saving time from a regular round trip flight, there's also the factor that flights out of Venice tend to leave VERY early in the morning. Getting to the Venice airport at that hour is time consuming, expensive, or both. It's easier to get a flight out of Rome leaving later, and even if you have an early flight out of Rome, it's easier and cheaper to get to the Rome airport early. Plus, Venice is a good place to get over jetlag. So, starting in Venice and ending in Rome works very well.
If you say which US airports are the ones you can depart from, we can help with flight options. To search yourself, use the "multi city" or "multiple destinations" function of airline search pages like Kayak, Skyscanner, Google Flights, etc, rather than the "one way" or "round trip" options.
To get between these cities, the trains will be cheaper if you buy in advance and can accept non-refundable and non-exchangeable tickets. On the other hand, if you don't mind full fare (which isn't too exorbitant on these routes), you can wait until you get to Italy to buy your train tickets. However, with a short trip like yours, you shouldn't have any difficulty knowing in advance which trains you'll want to take.