Trenitalia site for may 6 or around, Florence (santa maria novella) to la spezia,- 2 trains, about 6 hours with stop in Milan for one.
I know there are direct trains,should be about 2 hours, what am I doing incorrectly??
Well, it's direct not "non stop", but the Regionale train is what you're looking for. It takes 2 1/2 hrs, with about sixteen stops, but no change of train. Those aren't available more than seven days in advance.
You can always get tickets at the train station in Florence when you get into town.
Most of the trains will go through Pisa and then up to La Spezia.
Choose a date in the next seven days from today. Preferably the same day of the week as May 6. Regional trains timetable is available online only up to seven days ahead. I haven't checked but I doubt there are too many non stops from Florence to Spezia. Most options will require a change in Pisa.