I find that thetrainline.com is slightly more user friendly for booking tickets than trenitalia. They do not add a booking fee. In March it is unlikely that trains will sell out so if flexibility is important you can just wait till you are there, but the lower priced tickets will almost certainly not be available if you wait. And as mentioned, ticket lines at Italian rail stations can be really slow. I've waited in line for over an hour when there were only 4 people in front of me! So you just have to balance price, waiting time, flexibility to decide if you want to book in advance.
I also have never chosen first class. Once I accidentally ended up in first class (was booking at last minute) and I'd been in the seat for some time before I realized I was in first class. Hardly any difference.