Trenitalia website is the official website and where you should find the most complete list of train timetables. A quick search with date for next week shows several in the morning. There are also several trains from noon onward that have you back before 5pm.
As you can see, only the 8.16am train takes less than 2 hrs (train change in Pisa Centrale) - almost all the others take about 3 hours. If you want the comfort and speed of a private car, you can most definitely look into that as well. It will give you the security as well as ease of being met at the port and dropped off there, and not have to worry about getting to the train station and back on your own.
I just met Walter Occhini the other day and he's based along the Tuscan coast, so specializes in these types of trips from La Spezia and Livorno to Florence. You could get in touch with him and ask for a quote, at least you have something to compare to. His website is
www.privatedriveritaly.com
Let him know Lourdes from DiscoverTuscany sent you his way!
