re: Best way (cheapest) to book a flight to Italy?

After all of my flights, I still have never done a connection switching airlines (not talking about skyteam/oneworld alliance partners, but rather budget to popular airline) in an international destination

I did it this summer, and I wouldn't recommend it unless you get a fantastic deal. I flew Delta to Dublin round trip, then Aer Lingus to London from there round trip. HUGE PITA. Going was super easy and normal, coming back was a HUGE PITA. For one, for most international flights you have to check in at the desk because they check your passport. Huge airports that's a PITA because your regional terminal may be different than your international flight terminal, and you may not be able to do it at all until you are in the airport you fly out of. Saved me $400-500 per ticket so probably worth it. But I wouldn't do it for $100-200.

I would not recommend getting a connecting flight in Europe unless 100% necessary. Try to fly from the USA to your destination in Europe. We didn't get into London until about 1, by the time we got our rental car and got to our destination it was 3. So for one, it costs you almost a half day extra each way. Where as, a direct flight into London from the USA gets in around 9am like we got into Dublin at 8:30. But then had another flight at 11:30. Last thing you want is a connecting flight after a cross-ocean flight. Sucks