Connecting from an international to a domestic flight in Rome (FCO)

You'll have to pick up your bags and go through passport control. From there it really is easy. I'm assuming your next flight is from another terminal, which just happens to be only a few minute walk from outside international arrivals (actually, right next door - exit and turn left). You'll have to check in at ITA and they'll take your luggage. I just did this and found the walk helped me get over the cramped legs from the long flight. The ITA lounge is worthwhile, imho.

Thank you so much for this info. One more thing. We will have several hours layover and want to access a lounge in Terminal #1 prior to the flight. They do not allow FLIGHT CHECK IN more that 2-3 hours before the flight. Will we be able to CHECK OUR LUGGAGE before going thru security and then go thru security and go to the lounge? Then within 2-3 hours before the flight we can actually check in and get our boarding passes.

Hmmm, you got me on that one! I didn't have any problem getting to the lounge (after check in and dropping off my luggage). I don't recall a time schedule for checking in - I had more than 3 hours at the lounge, which is why I bought a pass for it. I arrived at FCO at 9:30 and my next flight was at 2:00. Can I suggest calling the airline (USA office was very helpful) and see what they suggest? It's not really a connecting flight but it is the same airline.
Also, I do remember the lounge was a long walk through lots of duty-free stores and difficult to find . . . but worth it. Let me know how it goes!

Is the flight to Bari on ITA or someone else? If ITA, contact them and see if they would be willing to check through your bags from Miami.

Yes, both on ITA. I will,call them thanks