george,
The first segment from Monopoli to Bari will likely be via Regionale train so you won't have seat reservations, and can sit anywhere you can find a seat. The final segment to Rome will be via high speed Freccia train and for that one you will have an assigned Carrozza (coach no.) and Posti (seats). Note that the tickets for the Freccia train are specific to train, date and departure time, and can only be used for the one train listed on the ticket.
According to The Man In Seat 61, the rules for Regionale tickets changed in August 2023....
"Ticketing for regional trains
The old system of having to stamp your ticket in a validator machine is gone, as is the system of tickets being good for any train in a 4-hour time slot.
From 5 August 2023, tickets for regional trains bought online or in an app are only good for a specific train.
You can change the date & time of your train free of charge using the Trenitalia app or website, as many times as you like, up to 23:59 the day before departure. It's also refundable up to that time. On the day of travel, you can change the time of your train as many times as you like, free of charge.
Before boarding the train you must 'check in' online using the app or the website link provided, this validates the ticket for use on that train, your ticket is then considered used and no further changes can be made.
Remember you'll need internet access on your phone for this, if you can't rely on that, buy at the station at least 5 minutes before departure."