If you want to travel in Italy do it soon, before the Islamist rub out all of the historic artifacts
Ten must see places near Rome
Well, kinda near Rome anyway.
I discovered Civita de Bagnoregio back before it was discovered, and rented an apartment from the owner. Wife, I, another couple and their newborn all stayed there for a week. It was a lot of fun. There is a short story about Civita, and a video at the link below.
Incredible Yemen village
When we fly into Rome, we always plan to spend the first 4-5 hours post flight at Ostia Antica. There is a little village right outside the grounds where you can grab a bite to eat, and the grounds are interesting, and offer plenty of room to stretch the legs after the interminably long flight. It is more interesting than many more sought after sights.
We also never stay in Rome for our first stop, often staying at a farm, B&B or the like up nearer to Civita de Bagnoregio.
The other sights are fine, but if you get off the trail, and head west out of Orvieto there are many interesting towns, and the baths at Saturnia are beautiful. Find something that not everyone and their cousin is writing about, park the car, and wander about, sit and watch a bocce match with some old timers. Discuss technique. It doesn't much matter if you speak Italian, if you just try, and use some good body language, they will gather around, agree, argue, face palm, and laugh with you, and sometimes at your language faux pas. Always ask people about what you see, and talk to everyone, we have spent many an enjoyable afternoon in the garden of a shop keeper after asking about something.
I once got into a discussion about rise gelato with a young woman behind the counter in the gelateria just behind the Duomo in Orvieto. After a few minutes she took us all down to the garden in the back of the home/shop, and showed us a variety of grapes, one plant had grapes as big as eggs. We sat, ate grapes, and drank wine for a couple of hours, while we discussed pretty much nothing - la dolce far niente.
When in Italy, la dolce far niente is my complete goal. It has worked so well that once while in the tiny winery in Civita with a friend from the US (he, wife, and child were stationed at the NATO base in Napoli), a tourist entered, wandered around, heard us speaking in Italian, and asked us a long list of questions about the winery, in very broken Italian, we answered as best we could, toured him around the little place, and showed him the cellar. As he was leaving I said, "Goodbye, come back and see us again." I think my "flawless" english surprised him.