On the two trips we've taken, we've started by booking flights and worked our way in from there. Get the air, then worry about the rest. Don't know when exactly you're flying in June, but it looks like the only non-stop is the 5:05 PM out of Dulles, arriving at Rome Fiumicino at 8:10 the following morning. From there you get over to the Roma Termini train station via the Leonardo Express train right there at the airport. Then you can take the train north to wherever, be it Cinque Terre, Siena or Florence, all doable in a few hours, so you'd in theory be arriving at your destination by late afternoon, or maybe even earlier.
The non-stop flight home (I just plugged in 6/15 & 6/29 for the heck of it) leaves Fiumicino at 10:50 AM (returning to Dulles at 3:05 PM), so you will want to stay in Rome the night before. In fact, it's almost a shame to be in Rome and not see anything, so I would suggest staying 2 or even 3 nights there at the end of your trip. Lot of things you could accomplish, even in that short time. We did that on our initial trip abroad--finished up in Rome, stayed near Campo de' Fiori and utilized the Hop-On-Hop-Off bus to--albeit somewhat quickly--see a few of the sights.
Here's the thing--I know you want non-stop flights--and from what I described above it's certainly doable--but I am sure you're finding out there are very few non-stops to Italy these days from the US & they're much more expensive. You could end up paying as much as $1000 more for the two of you. And United--probably your non-stop carrier--is uh...less than pleasing with their accommodations in flight, and I'm being kind when I say that after a Chicago to Paris flight last month. But maybe you have an upgrade to business class or better, which we looked longingly at as nirvana on our last trip! Lufthansa is a pleasure in comparison, but they don't do non-stops from Italy--we did a non-stop return from Munich after flying in from Naples. And as stated above, the flexibility of open-jaw flights (into one airport, out of another) opens up many more travel possibilities.
Have fun hammering down your plans--we're here to help you, and we'll gladly take your trip for you, but only if you'll pay for it! :)