Rail Europe in Italy

Rail Europe is a US agency and charges more.
Use trenitalia.com

For travel between large cities on high-speed trains, Italo is another option. Trenitalia is the largest rail company and serves both large and small destinations using a variety of train types. Italo is much smaller and more selective in its high-speed routes. Both also now offer bus links to certain destinations not serviced or difficult to service by train. You mentioned "flights" above, but I suspect you meant trains.

You can purchase full price tickets either in advance or on day of travel that can be changed up to around departure time (check Trenitalia and Italo rules for exact details). Tickets purchased in advance at a discount come with restrictions, such as no refunds or exchanges for the cheapest fares. If you plan to buy full fare tickets and your exact travel days/times aren't set, you can just as easily buy them at a train station or travel agency once in Italy. I always buy my tickets directly from the train companies rather than from a reseller.

https://www.thetrainline.com/ is a reseller that has the exact same prices as the national network. People sometimes have trouble using the national network websites (either they won't take a US credit card, or they redirect you to the more expensive Rail Europe reseller that also doesn't show all trains & times), but trainline has been easy to use for us.

As for changing trains, the cheaper tickets have more restrictions.

"I am planning several train (Rail Europe) connections in Italy" - to be sure you are not misinformed. There is no such thing as a Rail Europe connection.
Rail Europe is not a railway company. It does not in any way, own, run or operate any trains. It is a ticket reseller (travel agent) in North America.
Trains in Italy are run mainly be Trenitalia (the national railway company), also by Italtreno, who compete on some routes, and some local companies.

Tickets for High Speed trains are cheaper if bought in advance. Tickets for local trains it makes no difference.
Best place to buy is direct from the rail company: https://www.trenitalia.com/en.html

For a beginners guide to trains in Italy see here: https://www.seat61.com/Italy-trains.htm

"How easy is it to change to earlier or later flights if you already have tickets in hand?" - assuming you mean trains and not flights, it isn't. Once you buy a ticket, it has the train details on it (date, time, train No., coach and seat numbers). It is only valid for that seat on that train. If you want flexibility, don't buy until you arrive at the station, ut then you pay full price, no discounts.

Several errors or misconceptions in your post. First, Rail Europe is simply a reseller of train tickets. They don't run any trains, anywhere. And they don't sell tickets for all trains. You're better off using the actual train websites to check times and prices:

https://www.trenitalia.com/en.html

https://www.italotreno.it/en

I think you mean train trips, not connections or flights.

As mentioned, if you have advance purchased a full fare ticket you can change the ticket at the station for a different train. You can't do that with the discounted tickets. But since there is little point on purchasing advance tickets except for the discount, then you might as well just buy your ticket at the station, either from a machine or a ticket counter. Be sure to allow plenty of time for this as there can be lines.

It sounds like you are totally new to trains in Europe. I suggest brushing get up on the subject by spending get some time on this site: https://www.seat61.com/

Thank you so much for the responses, you’ve all taught me so much. My first time to Italy and I appreciate all the valuable information.