USA to Rome to Bari

When you land in Rome, you will go through immigration.

If you fly onto Bari, you will go through customs in Bari.

If you take the train to Bari, you will go through customs in Rome.

between the two I'd lean towards flying instead of the train. Your plane may be delayed into FCO and time thru customs is unpredictable, then you need to take the Leonardo shuttle to Roma Termini in time to catch the train to Bari (and there aren't that many per day anyway).

But if this was my trip I'd spend the nite in Rome on arrival and take the train out the next day.

edit: with respect to the post below, I was assuming this was all one ticket. If you're buying a separate ticket on RyanAir or ITA then things become a bit dicy. You'll have a lot of wasted time sitting in the airport, and even then there's no guarantee because in the months before your flight they may change the schedule.

My question is once we get off the plane can we just go directly to the gate to Bari?

No

You will need to go through immigration (passport control) upon arrival. As it sounds like you are contemplating separate bookings, you would need to retrieve any checked bags, then go and check them in and go back through security.

If all you have is carry on, you will have to be sure it conforms to the limits for FCO-BRI flight, or be forced to check it.

As a separately booked sequence of flights, anything less than 3 hrs between, is a risky proposition.

ITA air shows a single ticket solution for ORD-BRI with a 2hr layover at FCO

Turkish Air has a 16 hr solution via Istanbul ( ORD-IST-BRI)

Depending when your planned flight lands, it could be possible to take the train, but that involves almost 5 hrs train time, which after an overnight flight may not be appealing.

Wow, thank you for all your replies. Very helpful.