If you rented a villa in the countryside outside Florence you will need a rental car.
Actually since there are 6 adults and the cost of renting a van is equal to the cost of renting 3 small cars, I suggest that you rent 2 economy size cars.
If you rely on a private driver on a van for your trips to/from the villa you will spend not hundreds but thousands of euro.
You can rent 2 economy/compact size cars for less than $200/week each, fully insured. Contact a consolidator for booking:
www.autoeurope.com
www.kemwel.com
(both the above are part of the same company, but check prices with both).
From Florence to La Spezia there are many trains. Check schedule on:
www.trenitalia.com
The name of the main station in Florence is Firenze Santa Maria Novella.
The trip to La Spezia takes over 2 hours and in most cases requires a train change in Pisa.
Once in La Spezia Centrale, take a taxi, the port is only 10 min drive away.
However if you follow my advice to rent cars, it is more convenient to drive from Tuscany to La Spezia and return the rental car there. Most rental car companies are located near or at the port of La Spezia.
