Train from Rome to La Spezia - stop at Pisa?

Not on most trains from Rome to La Spezia Centrale. These trains look like they are ICPlus trains. You will most likely be on a train that has a definite train number and departure time. You will also have seat reservations on that particular train. These trains all stop at Pisa on the way to La Spezia Centrale. The problem is - you have to stay on the train as part of that ticket. If you get off a Pisa and the train departs, your seat reservation goes with it.

To resolve this problem, you book a ticket from Rome to Pisa and a separate ticket from Pisa to La Spezia Centrale. The first ticket can be on a train that requires seat reservations and firm departure times. However, if you book the second ticket on a regional train (R), no seat reservations are permitted and there are no departure times on the ticket. It's an open ticket permitting you to jump on any regional train withing 60 days of purchase. This permits you to stay in Pisa as long as you wish. Your train travel times could be a bit longer on these regional trains as they make more stops.

So the purpose is to go to Rome to Cinque Terre and then onto Florene, stopping sometime in Pisa to see the sights there... would it make more sense to do Rome to La Spezia (the do Cinque Terrie), and then go to Florence, stopping of in Pisa on the way?

Stacey, you first words were Rome to La Spezia. I assume that you were headed to the CT. Most of the train runs go directly west from Rome to the west coast of Italy and then north to La Spezia where you will change to the local train for the remaining very short ride into the CT towns. Do you wish to visit Florence before your visit to the CT? Or, will you be going to Florence from the CT? In either case, you can stop off in Pisa. You'll be stopping there in both directions. As mentioned above, if you are on a train that requires seat reservations (i.e. ICPlus, EuroCity), you will need a train ticket that ends in Pisa and another that goes from Pisa to Florence. If you are on an R train, which most from the CT to Florence are, you can get off at Pisa and get on a later train as long as you do this within 6 hours from the time you validate your ticket.

As a side note, you will have a separate ticket for each different train you are on. For example, it is possible to book a trip from Florence to Vernazza and have three different tickets as some runs require train changes at Pisa as well as La Spezia Centrale. It's just the way it works.

If you want to see sample Trenitalia train schedules, I can link them. Just let me know.

I went the other way, from Florence to La Spezia (then on to Milan). I bought a ticket to Pisa, dropped my luggage at the storage room, spent 1/2 day seeing the leaning tower, church & baptistry, then took the train from Pisa to La Spezia, then the local train to Riomaggiore. Not a problem, since I had to change trains in Pisa anyway to get from Florence to La Spezia, as I recall.

Chani's trip is typical. From Florence to the CT, you will most likely change trains in Pisa or La Spezia and sometimes both. As Chani notes, there is a staffed luggage storage area at the Pisa train station. We've used it. As you face the tracks, turn left (south) and walk down to the last buildings. Turn left in between the last two buildings and you will see the staffed storage area. Your luggage is stored behind the counter on shelves. When we were there, they made a copy of your passport picture and placed it with the luggage. To get your luggage back, you had to show your passport. 3Euros per bag. Just be careful to note the operating hours.