KS,
There are several direct trains from La Spezia Centrale to Firenze (various stations), including departures at 07:44, 13:35,15:40 & 17:40. Travel time ranges from 1H:35M to 2H:32M, and different classes of train are used on that route. I used an arbitrary date of 9 June to check, and I doubt the new schedules after that will change much. Could you clarify if the most important factor for you is travel time or number of changes? Which of the five Cinque Terre towns are you staying in?
In that situation, I'd probably use a departure from La Spezia Centrale at 10:06, arriving Firenze SMN at 12:00 (time 1H:54M, one change at Pisa Centrale). That would allow you to still have a nice breakfast at your C.T. hotel, and a leisurely trip down to La Spezia (only takes a few minutes). That gets you to Florence at a good time of day to check into your hotel, and hopefully by that time your room has been serviced and you can check in straight away. After check in, have a relaxing lunch and a brief walkabout to familiarize yourself with Florence. That still leaves lots of time to get to your dinner reservation.
That's a relatively short route so you likely won't save much with advance purchase tickets, especially as some of the trains you'll be using are Regionale as there's no discount for those. I'd suggest checking the Trenitalia schedules and then buying your tickets the day prior to travel so you don't have to deal with that when you're loaded down with luggage. One important point to mention is that if using locally purchased Regionale tickets, you MUST validate these prior to boarding the train on the day of travel or risk hefty fines, which will be collected on the spot! If one of the trains you'll be using is an Intercity or Freccia, these have compulsory seat reservations and are specific to train, date and departure time, so the tickets for those don't have to be validated.