Where to fly into?

Flights are not always cheaper to Rome. You could start your research using www.kayak.com. Besides Rome(FCO), look at Milan (MXP), Pisa (PSA), Florence (FLR) and Naples(NAP) as possible starting and/or returning cities. If you can't find schedules & prices from MSP that you like, consider flying out of O'Hare which has many more options to choose from(the 7 hr drive down to Chicago may be worth it).

You should also look at open jaw tickets. Often they can be cheaper in total cost. Into Rome, home from Milan or something similar.

I would look at flying into Pisa and out of Rome. From Pisa you could go to Cinque Terre first, then Florence, through Rome to the Amalfi Coast, then back to Rome for your flight out. A nice thing about the Pisa airport is that there is a train station right there, so you can connect directly to the train system without a bus or taxi trip into the city. Milan may offer good fares, but it is not close to any of your intended stops.

I do not know anything about the comparitive prices. But the Florence airport is nice & small. There is a bus stop right outside the Florence airport that has a shuttle that runs about every 30 minutes between the airport and the historic part of Florence (right next to the train station). It is the least stressful airport I have used. So if I have a choice that is the one I pick to arrive or leave.

For your itinerary, flying into Pisa or Florence and back from Naples, or vice versa, would work well. You can pay about $100 more than roundtrip Rome for open jaws (about what you'll save on train tickets by not back tracking) and still break even. You may also want to consider the convenience of open jaws and decide how much more than $100 you'd be willing to pay for the extra non-travel time you'll get.