Train from Bolzano to Verona to Rome

They are very unlikely to be full unless you travel on the first weekend in August or the long weekend around August 15, which is a holiday. Even then, the only trains that might be full are the non-stop Bolzano-Rome (Frecciargento), but if you are willing to change in Verona or Bologna, you'll find plenty of solutions with no chance of being full on the day of travel. If however you are sure of your travel plans and you can find an Economy or Supereconomy ticket (they usually go for sale 120 days prior and sell out fast), maybe you might consider pre-purchase. Take into account however they aren't refundable and changes are restricted too, therefore you must be certain of your travel plans.
Said this, on the day of travel you must purchase a reserved seat on Freccia or InterCity trains. You can't get on board on any train unless it's a regionale. Regionale tickets must be validated at the machine prior to boarding.

Mary, Could you clarify if you're doing the trips in segments (ie: are you stopping in Verona for a night or two on your way to Rome). There are a variety of trains used on the Bolzano - Verona route (Regionale, EuroCity, Regionale Veloce). If you want to "grab a train at any time", buy a ticket for a Regionale when you arrive in Bolzano, and then just validate the ticket prior to boarding the train on the day of travel. BE SURE that the train you're boarding is not a EuroCity or other train that requires compulsory reservations, as without a valid reservation you'll likely be fined on the spot! Travel time via Regionale is ~2H:15M and via EuroCity ~1H:25M. You can check all the train details using the Trenitalia website. Happy travels!