Great advice above!
March 24-28 Rome
Will the baby have his or her own carseat on the plane? Our baby granddaughter did and in addition to its being safer and then later essential in the car (actually a van), it meant that she and her parents could get some sleep. And it snapped into her stroller. I felt iffy about the stroller and our sons never had one, but I had to admit it was really useful for walking around and she napped during restaurant lunches. Our grandson, like me, could not sleep sitting up on the plane, so our son stood up for hours and let him sleep stretched out on two seats. He had a carseat, too, but it was checked as baggage.
It is not, not, not fun to lug around a 2nd carseat in train stations, etc., even with four adults to carry stuff, so you might be able to rent one along with the rented car instead. We also had a travel bed for our granddaughter, but if you rent the right apartments, with a baby bed, it shouldn’t be needed.
As you know, you will all be jet-lagged. Adults can cope with this, but if your kids are like our grandchildren, they will be awake at night and, uh, not coping. Make a plan for this, perhaps a 2-bedroom apartment and each parent sleeps in the same room with one child. I could tell you about a perfect such apartment, but it’s near Termini, not in Ponte.
March 29-April 3 - Train to Bari
There are fast and slow trains from Rome to Bari. I like to look up trains on Google Maps to see all the times, then use the train app to check Google and buy your tickets. Since you are staying in Rome after your flight, and can be pretty sure to catch a particular train, you could buy the tickets ahead of time.
One alternative to the Rome–Bari train is the Rome–Foggia train, and then rent a car in Foggia. We did this once. Shorter train trip and easy to get to the rental car office. I don’t know where the rental agency is arriving in Bari by train — we’ve only flown into Bari and rented at the airport. We’ve also rented cars in Lecce and in Brindisi. We like Europcar, but on our latest trip to Puglia, we rented from Hertz at Termini (train was sold out so we drove from Rome to Lecce).
On the trip with our grandchildren, we stayed in Ostuni (in the oldest part) for a week and in a trullo near Cisternino for a week. On another trip we stayed at Masseria Aprile outside of Locorotondo, also in a trullo. All good. The 4-year-old loved Ostuni, Lecce, throwing rocks into the water on the beach at Polignano a Mare, the train museum in Lecce, and running around in various piazze in various towns. He really loved Alberobello. He loved all sorts of things about just being in Italy, like how toilets flushed and how some sinks had foot pedals. He was great in restaurants as long as one of us took him outside to explore and run around until the food came and he had things to do at the table when he was done eating. His sister ate her very first solid food on this trip — a ripe peach which she gobbled with extreme delight.
It was fun to take a baby to Italy. She was greatly admired wherever we went. I don’t know about New Zealand, but in America you don’t get old men and school-age boys coming over to admire a baby, usually just women and girls. All the restaurants were so welcoming and accommodating, too.
