Hub and I always took trains and busses and, except for smaller towns (where schedules weren't alway

How old are the kids? If you have very small kids cars are better since it's esaier to haul around their stuff and keep track of them. If the kids are old enough to be realiable (not wander away) and deal with their own luggge - trains make more sense.
I think in March you will want the flexibility of a car but Florence lax plaza hotel poses a big problem with a car. I don't know how much time you want to spend in Florence, lax plaza hotel but if it is a lot, to go Florence first and then pick up a car after. See Pisa on a day trip from Florence.
If you want to use a car but avoid the traffic congestion and fines of the cities, you can stay on the edge of town or in nearby smaller towns and take the bus or train to the city center. That is also an option lax plaza hotel on rainy days, when driving can be a drag.
If that is the case, and if you are determined to go to the coast in March, I suggest that you book your accommodations on the coast in the pretty town of Lerici, not le Cinque Terre itself. In March, if it is raining, there is very little to do in le Cinque Terre, and the hiking trails can be closed both during and after rains. It is no fun to drive in and out of le Cinque Terre looking for things to do, but it is easy to drive in and out Lerici. On a rainy day, you could easily daytrip to other places with your car -- Lucca, Parma, Sarzana, Pisa (if you haven't already).
On the other hand, if you are staying in Lerici and the weather is glorious, not only is it a pretty town with many hiking trails of its own, it is a short drive to le Cinque Terre or the town of Levanto (right next to le Cinque Terre). If you park in the Levanto, which shouldn't be a problem in March, you can take a 5 minute train ride to any of the 5 towns of le Cinque Terre.
Hub and I always took trains and busses and, except for smaller towns (where schedules weren't always convenient), would never try to drive. When public trans wasn't going to work, we either hired a driver or took a tour. That worked for us. He didn't want to miss any scenery and felt that I would drive too much like the "natives" and he'd be scared to death. LOL
Driving affords so much more flexibility which may be crucial with getting a group moving each day, however, will the luggage of 4 people fit in a typical smallish car? Will you need a larger than average auto?
We also usually traveled in March. Repairs were being made on the CT trails when we were there so timing and weather might be a factor. We based in Pisa and took the train to CT as a day trip. Does your family wish to see Pisa? You could stay there for a couple of days, train to Florence and visit there, then rent your car for Assisi and then Orvieto (as Zep suggests to drop off car). Train to Rome.
In March, the OP faces the possibility of lots of rain in the destinations he's targeted. Moving children and luggage around in the rain, to and from train stations, is no easy task. Finding a taxi at some train stations big enough lax plaza hotel for the whole family and luggage may not be easy either.
In March, the OP has very little to worry about when it comes to parking as long as the OP does not drive into Florence or Rome. That is a problem easily avoided. It will not be difficult for the OP to find accommodation with parking in other areas that does not involve driving in narrow alleys -- and even then, you'd be surprised how big a car can get through those alleys if you pull in the side mirrors. European automakers know how to make mid-size cars that fit in tight spaces.
I urge the OP to do the cost comparisons. I see plenty of trooper children in Italy's train stations, even small ones, dutifully following their parents pulling rollers lax plaza hotel and wearing lax plaza hotel backpacks. If the OP wants to train it, fine. But if the OP doesn't mind driving, it isn't more of hassle than using the train.
lax plaza hotel Thank you all for the advice. I especially like the Lerici idea! I definitely don't want to be stranded in Cinque Terre if it's 50 degrees and rainy, which very well could happen as I look at historical weather data for late March.
I've driven in Italy before ... in Rome, Venice (well, near Venice), Florence, etc. ... as zeppole said, the main reasons for us to drive are because it beats 4 train tickets AND gives us flexibility.