Susan, to add to the previous posts, these would be my suggestions:
Rome to Siena - Bus would probably be the best option if time isn't a factor, as it will drop you in the centre of town (as opposed to the bottom of the hill, where you'll have to take a Taxi to get into town). Check www.sena.it for Bus information.
Siena to Florence - definitely Bus! The Bus depot in Florence is almost right next to Firenze S.M.N. station, and it's a very easy trip.
Siena to Venice - do you mean Florence to Venice? If that's correct then definitely Train! There are several trains each day, and the travel time from Firenze SMN is about 2H 40 M. Note that reservations are often compulsory on that route. When you arrive at Venezia Santa Lucia, walk out through the glass doors, down the steps and you'll be at the canal. Check Rick's books for which Vaporetti are the best (No. 1?).
I didn't address the question of "cheaper or faster", but rather focussed on the easiest method, regardless of cost. I don't think the cost will be a factor for these three short journeys.