These are the Trenitalia conditions for taking a pet according to its sizes: 

• Small Pets: You may travel with up to one pet for free in first and second class. However, they need their own ticket and must travel in a carrier that measures no more than 70 x 30 x 50cm. To issue the free ticket, please contact us through the contact form. 
• Larger dogs: If you purchase a ticket, you may be able to travel with your dog on a leash and wearing a muzzle. To make this happen, please contact us through the contact form. 

You must have a registration certificate and a health card for your pet (except guide dogs), or a "pet passport" to replace these documents. You may be asked to leave the train if you do not have this information.