Bus versus train travel

Don't know about your Rome-??-Florence leg, but Florence-Venice should be by train. As an earlier poster noted, Italy has no real long-distance bus service like Spain or the UK. Buses do serve regional areas, so your section from ?? to Florence might be doable by bus. But clarify the town's name, then check with the Italian railway network first at http://www.trenitalia.com/en/index.html

I think you will find the train the cheapest and most convenient way to go. It will be the most direct route and I'm sure faster than the bus.

Definitely train. Also, into and out of the big 3 cities you mentioned, I would make sure to have a specific seat reserved, because those trains are crowded and you might end up standing for a while.

I wouldn't expend much energy looking for the 'cheap' way between the major cities in Italy. If you've got no wheels, you're not fluent in Italian & are a tourist - make use of the tried & true methods (ie. train).

BTW, are you referring to Vairano Caianello??? It's a regional line so it's not going to be expensive anyways. But ask at your hotel/hostel in Rome. Locals know.

I agree that the connections between large cities are probably best by train, the only exception might be if the two cities are not more or less directly connected by rail (routed out of the way through another stop, look at a rail map.) then check the bus. For smaller towns, bus may be the best option, again because train service may route you out of the way or connections delayed due to infrequent service. Start by laying out your destinations in Trenitalia and get travel times, then compare that with a map program to get drive times. If the drive time is much less, then consider bus. Like others have said, cost should not be much different, but time may be.