Train from Naples to La Spezia Train Station

Liz, Perhaps the easiest solution would be to travel via Freccia from Napoli Centrale to Roma Termini, and then connect with one of the direct trains to La Spezia Centrale. There are only a few trips each day with the direct trains, so you'd have to time your departure from Naples accordingly. My preference would be a departure from Roma Termini at 09:40, arriving La Spezia Centrale at 13:46. There are numerous other trains on that route but they have multiple changes (always easier to use a direct train). You can purchase tickets in advance on the Trenitalia website for the premium trains such as the Freccia (which will save money if you can get "Super Economy" tickets), but keep in mind that will "lock" you into a specific train and departure time. If you miss that train for any reason, the tickets are worthless and you'd have to buy more at full price. One other important note - reservations are compulsory on the premium trains, and will be included with P-P tickets. These are specific to a particular train and departure time, so you can't just board any train going in the direction you're travelling. DO NOT be caught without valid reservations for the train you're riding on, as you'll be fined on the spot and it's not cheap! If any of your rail trips are on Regionale trains (which don't require reservations), you MUST validate your tickets prior to boarding the train on the day of travel, or again fined on the spot! Happy travels!

From Naples to La Spezia you will need to change trains at least once. Use www.trenitalia.com to see schedule. Enter from NAPOLI CENTRALE to LA SPEZIA CENTRALE. Choose a date within the next week.
For time enter 00 (so you see all trains). You can book in advance on the same website. Buying online in advance might give you a chance to snatch a discounted ticket (Economy or SuperEconomy). However discounted tickets come with restrictions, they lock you in a specific train and time, and you are not allowed refunds or changes if you change your mind later. If all they have left is full fare tickets (called BASE) for your day of travel, then there is no advantage in buying ahead. You can buy them on the spot, since they rarely run out of seats. At most you'll have to wait for the next train 30 min later.

From a very quick look at the schedule, looks like most connects through Rome and a few through Florence. So no direct. You can book all the tickets at the same time. The ONLY advantage to reserving in advance is the availability of discount tickets. But the discount ticket locks you into a schedule with no refund or exchange depending on type of discount ticket. All trains in Italy except for the Regionale trains require a reservation. The Regionale trains are not discounted and are open seating. Just get on and go. You need to learn how to use trenitalia.com to answer your scheduling questions.