Traveling from Vicenza to Naples by train via Padova

With the children, I suggest taking an earlier train for the first leg from Vicenza to Padova so that you can allow more connection time. Unreserved Regional trains bring the choices to about 4 per hour.

There are many variables unanswered here.
Any other adults with you ?
And luggage?
Will the kids be strapped into stroller, and if so is it double width or length?

Padova is a relatively complex station. Relatively meaning if you from the USA it will seem complex, europeans used to rail travel will see it as OK. Unfortunately Trenitalia don't show platform numbers in their advance timetable. But looking at it's layout I'm guessing that the onward train will depart from the first platform which the main building opens onto, and the inbound train will be at least a couple of platforms over. In which case you have a dash through the underpass, of which there only seems to be one, right in the middle of the length.

The big question we can't answer is will the trains be on time (they were this morning).

FR9711 is scheduled to arrive at track no. 5 at 10:42
FR 9419 is scheduled to arrive at track no. 1 at 10:51 (and departs at 10:53)
So you have 7 min for the detraining and transfer to the 9419.
As with airplane gates, occasionally these may be subject to change if some trains get delayed and scheduled tracks are occupied.
Assuming the scheduled tracks are unchanged, that means you need to take the stairs downstairs to the underpass to walk 2 platforms over. The first Platform serves track 1. The second Platform serves track 2 and 3. The third platform serves track 4 and 5. The signs indicate the track number so just walk up to the track levels from the underpass when you see track 1, which is actually the station building (the exit).
If the arriving train is on time, you can make it but..be sure you are the first one closest to the door ready to get off the train when the train arrives. Head to the SOTTOPASSAGGIO (underpass) and walk toward the main station building (track 1 is the first one closest to the main station building). If you are in front of the train before 10:52 you can make it. The doors close only when everybody is on board and the conductor gives the signal to the engineer.

If the first train is late and you miss the second one, Trenitalia will protect you even if you purchased a discounted ticket. Of course you must ask for a new ticket at the counter, you can't simply get on the next FR train going south.

Edit: Roberto checked the intended tracks without knowing the arrival and departure hours of the trains. Sir, have my like. Out of curiosity, did you first checked on viaggiatreno.it and then on RFI or there is an easier way?

Thank you all for your quick responses. I will be traveling with three other adults, however two of them will be in charge of carrying car seats (needed for a rental car in Naples) and one in charge of the carry on luggage. I have not checked the viaggiatreno website, but I will. There is another train that leaves earlier and has more time available for the change, however it goes through Verona. I thought maybe Padova might be a little smaller and easier to switch between trains.

When you're searching for train schedules, there are ways to adjust the amount of transfer time you want. But you can also just search separately for the Vicenza-Padova schedule and see all the options.

There are two Regionale trains - one at 10:02, the other at 10:08 from Vicenza to Padova. They are less expensive than the Frecciarosa and will allow more time to change trains. When you arrive at Padova, check to confirm what track you must go to. Each platform has an elevator and I suggest with young children that this will be easier and safer than taking the stairs. The elevators take you down to the tunnel that spans all the tracks so when you get off, simply walk to the new track and take the elevator up to the platform.