Vicki, first, all train runs from the CT to Venice do not go through Florence. Many runs go through Parma and Bologna and do not go to Florence. As metioned above, the 7:35am from Monterosso to Venice does this.
Second, the Eurostar-class (ES) train is the fastest and most expensive. The InterCity (IC) train is more regional. It is less expensive and takes longer as it stops more often. The Regional (R) train is the local. It stops at every stop. It's the cheapest ride of all.
In the middle is the ICPlus train. Fewer stops but not as expensive as the ES. For your trip from Rome to the CT, many of the runs go west to the coast and then north to the CT. Others will take you to Florence where you change trains. The coast routes can be slightly faster.
Given you itinerary, you most likely would not be on a ES train if you take the coast route from Rome. Also, from the CT to Venice, you won't hit the ES train until Bologna. Your trip from Venice to Florence will most likely be on the Eurostar.