Help with planning a 5 week trip in April/May in Italy.

Most people will take the train down to Naples (about 150 mi.) and use buses around Sorrento area. As you said, parking can be difficult and driving can be hectic down there.
Why don't you catch the train back up to Rome and on to the C/T (through La Spezia) before going to the Florence area. And then take the train to Venice, which is about 150 miles from Florence. Parking in the big cities is expensive, and there's a chance you could get into the ZTL's in Florence. Venice is famous for their radar cameras issuing tickets--and don't ask me how I know.
If you want to explore Tuscany, rent a car in a city center as you're leaving Florence and turn it in as you're moving on to the next city.

If you are spending a week or so, I would think you have the luxury of time to enable you to make use of public transport. - Circumvesuviana local train as well as buses.

The only time you would want or need a car would be to explore towns and villages in Tuscany/Orvieto. For the rest, it will be more trouble than it's worth.

Sorrento is a good base, since it is connected by local train, buses and ferries to the entire area. You'll enjoy the drives to Ravello, Positano, Amalfi, much better on the bus - driving is difficult.

I agree, the best route is to start there, then Rome > Cinque Terre > Tuscany > Venice.

Do not go to the expense of renting a car until you get to the Tuscany portion of your itinerary. Trains and buses are super here.

thanks to all that gave suggestions. As we are elderly and can't drag suitcases easily from train to train or bus, we felt like a car, even with the inconvenience of parking, is still our best bet. We will do the Naples part of the trip with public transportation but then will get the car and head north. And don't suggest just carrying a back pack to make things lighter; we can't pack that lightly. We appreciate all the feedback.

Lots of folks here like to play down the benefit of having a car in Italy. For me, it was the ONLY way I enjoyed my two months. You can visit on your own timeline without trying to figure out train schedules, bus stops and the worry of "missing" that last departure. I drove from Milan to Florence, Venice, Rome, and all the little and big Tuscan and Umbria towns in between .... at MY pace. Being your trip is 5 weeks, check out Leasing rather than Renting. I used http://www.autoeurope.com/ absolutely a first-class outfit. Parking is easy to find outside the ZTL areas.

A car in Italy will not necessarily make dealing with luggage easier. That's because almost all Italian cities - of every size - have a ZTL, a central area in which no visitors' cars are permitted. So, you have to park outside this zone, and take a taxi or bus to your hotel. As long as you have to take a taxi to your car, you might as well be taking a taxi to a train station.

And in Venice proper, no cars can go at all - all motorized transit is on water. So, if you drive to Venice, you will be parking the car, then taking a water bus (vaporetto) or water taxi (very expensive) to your hotel.

disagree Harold. Parking outside a ZTL in towns that have them does not mean a taxi into the centro.. You can walk there from the parking lot. In all the places I visited the parking lots were close to where I wanted to go.

Yes Jim, you could walk from the parking lot. But can the OP walk with their luggage, given that they specified that handling luggage was hard for them? Only they can answer that question, of course. And of course, it depends on where the parking lot is in relation to their hotel.

I just wanted to point out that in (say) Madrid or Paris, everyone says not to drive, but you are allowed to pull your car right up to your hotel with your luggage. In most Italian cities, you aren't, unless you select a hotel outside the ZTL.

A car isn't going to solve luggage problems and is going to create an additional new set of problems. I'm short, old, and not strong, but I manage with a backpack and a 24" wheelie all over Europe. I usually take taxis from the train station to my hotel, because I don't want to have to pull a (for me) heavy suitcase around trying to find the hotel. Train stations are easy - they are either ground level or have elevators, escalators and/or ramps. If you can get through airports, you can manage most train stations. The hardest part is getting the suitcases up the 2-3 steps on the train, and there are usually younger, stronger people who will willingly help you. It shouldn't be harder than lifting them in and out of a car trunk - maybe easier.

Why don't you post your proposed itinerary and we can give you tips. Some of the train stations in the Cinque Terre require going up and down a flight of stairs. Venice can be difficult if you have to cross bridges to get from the vaporetto (water bus) stop to your hotel. Sorrento is arguably the best base for the Amalfi coast. To get there, you need to walk down one flight of stairs (there is an up escalator for the return) in Naples to the Circumvesuviana train to Sorrento. Then there are about 6 steps down to the elevator to ground level.