We were so busy sightseeing, even going to Rome Central Station to buy train tickets was the time we didn't have. Therefore, we decided to buy the tickets online via wifi in the hotel. Trenitalia's website in English version was actually easy to navigate and worked very well. It really saved us time.
I don't see a big price difference between buying online and buying in the train station. However, you save time buying online. There are some restrictions if you want to change the tickets bought online. We booked our departure time around 11am so it was not likely we would be late or anything.
We rode Frecciargento to Florence. It is a high-speed train owned by the Italian national train operator, Trenitalia. Frecciargento might not be as spacious and new as Spain's high-speed train Renfe, but the ride was smooth and fast enough. In one and half hours, we arrived in Florence.
See the picture of the train. JJ documented our trip religiously and never missed a shot. He even took pictures in the train station. I could not do that. The thought of pickpockets made me not want to be distracted from watching my belongings. Anyway, the exterior of the train needed some wash. They don't wash trains like Spanish do. Renfe was almost spotless like it was perfect.