ItaliaRail tickets
 

I've booked our train tickets from Rome to Tivoli - thinking now it might not have been necessary to prebook by oh well. When I booked I booked for Trastevere to Termini, Termini to Tivoli - I've realized it doesn't make much since to get on a bus (we will be staying near Bocca della Verita) - to Trastevere to get on a train to Termini.... I am assuming we can just skip that first part of the ticket and take the bus directly to Termini to catch the train there to Tivoli - BUT I don't want to assume I have any idea about Rome public transportation. Can anyone tell me if that is correct?

That would be fine. As you've realized these are all regional trains, and it really doesn't make sense to reserve these trains in advance.