"I found out that there is a train that runs from Florence to La Spezia"
Actually, there are 22 trains a day that run from Florence to La Spezia Centrale. Once there, 44 local trains run from La Spezia Centrale to Monterosso and beyond. Fewer trains go to the other 4 CT towns. North Americans are surprised by the enormity of the Italian train system.
If you were to look at the runs from Florence to Vernazza, you would see nine runs for the day from Florence. That's the way the Trenitalia schedule works it. It figures out the train changes, time of day you select and the trains available at the time.
Buses do not run from La Spezia to the CT towns with this volume of trains. Besides, the trains are literally buses on tracks. They are far faster and the stations are in the center of the towns.
You can easily go from Florence to La Spezia Centrale and wait for the next local into the CT towns. Or, better yet, book a ticket from Florence to the town you want. There are no direct trains. Depending on the run, you could change trains in Pisa, La Spezia or even both. However you do it, its about 3.0hrs from Florence to the CT.
CLICK HERE to see the tyical morning schedule.