Buongiorno,
though I would be tempted to tell you spend all your time in Tuscany, I can appreciate the thrill of seeing Rome for the first time - and in 11 days you should be able to dedicate two days to Rome. I sense from your enthusiasm that one day probably won't be enough.
But having said that remember that Tuscany is loaded with reminders of the Romans, for example check out Fiesole and Arezzo with their Roman amphitheaters. Next to Pisa you will find
Massaciuccoli with Roman ruins. And you can also look into the area
Frascole where yo will find good wine AND Roman ruins.
Buon Viaggio
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