Luggage

If it is in the summer you will need less clothes so it is possible that you can do it with large carry on back packs like a lot of travelers on this forum do (and my large carry on backpack actually has wheels) .

I have packed light and have planned to do laundry at times on trips after a few days that is always an option but of course you have to map that out which is not very hard in this era of google. And I never rent a car so you have even more flexibility.

The only trouble you would have rolling a bag in Italy is if you were to decide to walk a long way to get somewhere instead of taking a taxi or bus and there are cobble stones or hills.

On trains the only issues are lifting bags on and off the train. Like has already been said the issue with rollerbags is cobblestones and uneven sidewalks.

Well, whether you take a roller is up to you with pluses and minuses. However, I would not let cobblestone factor into the debate. I have rolled luggage over cobblestone from York, to Istanbul, to Cairo, to Delhi and, aside from noise, never had a problem with it.

I take a rolling carry one and it's been fine on all kinds of paved terrain. Pebbles slow it down. It has a top and side handle, so I could carry it as a suitcase.

If you bring 3 bags as you have suggested, you run the risk of them not fitting into the trunk of your predictably (and preferred) small rental car. Trust me, you would not want "anything" visible in the car if you were not within 10 feet of it.
Many of us travel well with one carry on size roller or backpack and smallish day bag or purse for 3 weeks or more with mix and match clothing choices and a willingness to do some hand washing at night once or twice a week.
I would strongly encourage you to size down ...

Some good points have been made and some bad ones.

Cobblestones are not a problem for most two wheeled rollers but can be an issue with a four wheel spinner.

If you carry a third, larger bag--something I don't suggest--you will be responsible for it. There are no porters on trains.

And, as stated earlier, you don't want to have any bag showing in a parked car. And cars are smaller in Europe.

You should be able to fit a week's worth of clothes in your backpacks. That means having to do laundry once or twice. It's not that big of a deal.

If a "large carry-on backpack" means something like Rick's Convertible Carry-On at 21" x 14" x 9", then in my opinion you can carry enough clothes for at least 2 weeks before you need to do laundry (not just one week). Our goal is for each person to have one bag of this size that you can easily handle alone. If one of you prefers wheels to a backpack, that's fine.

If you think you'll be shopping along the way, you can bring smaller, lighter, collapsible bags for those items, and/or your big bag might expand to fit them and be checked on the flight home.