We are training from Avignon to Cinque Terre, then to Pisa, Florence, Venice, Lake Como and Milan. Should we book trains as we go, or take the Eurail option? It will be 6 train days, but often short trips.
Definitely, compare individual point-point prices to travel with a railpass, but don't use RailEurope prices if at all avoidable. Those are just the prices at which they sell tichet, not the prices of ticket over there. They (tickets from Rail Europe) are usually much higher. There are less expensive trains that are not shown on RailEurope as well as promotional fares. If possible, get prices from the national rail companies (Deutsche Bahn, French Rail, Trenitalia, etc) over there.
Use Rick's books or a website like raileurope.com or similar to find the costs of the individual trips. Add those together and then compare to the rail pass price. This is what I did last fall and I found it was cheaper to buy individually. Also keep in mind that you may need reservations in addition to the rail pass price.
Point-to-point tickets for your rides in Italy will definitely be cheaper than a pass. Your best option will be to buy p-to-p tickets all the way.
Amica fares (20% off) may be available for some of your routes in Italy. Ask when you get to Cinque Terre. You can buy all your tickets there, if you wish.
I don't think a rail pass will do you much good. Italy is pretty cheap so long as you are willing to wait for the less expensive, slower train. Three years ago we bought tickets as we went, from the machines at the train stations (US credit card). The only secion I'd really check into in advance in case you need reservations is from Avignon to Genova (you can get your Cinque Terre train from there).