Train from Monterosso to Florence to Naples

Search your specific date at https://www.trenitalia.com/en.html to choose the trip with least connections at a time you prefer.
The connections will be fine--they do not take long and schedules are set up for you to have enough time (given reasonable expectations about your physical ability and amount of luggage you have).
No doubt it is a long day but if that is what you have chosen to do then it'll be fine--I'd be really active the day before to make it a rest day. I would not splash out for a private transfer.

There are no direct trains from Monterosso to Florence SMN so you'll need to change trains somewhere - probably Viareggio or Sarzana - but just be careful that you don't pick a time that has two changes because it can happen. It doesn't really matter which station - at least not enough to choose one trip over another . Just choose the time that works for you.

The train from Florence to Naples should be fast train with reserved seats and a specific time. Florence SMN is busy but it isn't large so transfers don't take that long since it's usually a matter of a few hundred yards or less. Get the Trenitalia app and it will allow you to check on trains in real time and so you should know what track you need to be at when you get off the train if your next train is close or in the station.

Whether an expensive car service is worth it can only be decided by you. It won't be much faster but it will be expensive. Trains are how practically everyone gets around Italy but only you can decide your time vs convenience vs money equation.

Have a great trip,
=Tod

There is actually one direct train from Monterosso to Napoli, but it leaves rather early at 5:18...

Don't worry about connections. The fastest trips you will see offered involve a local to La Spezia, and high speed trains from there onwards. If you miss a connection due to a delay go to the Freccia Service Desk (not the Ticket Office!) to have your ticket exchanged for the next available train.

Follow up: @WengenK is right and I had missed the 5:16am InterCity from Monterosso to Naples. I can't say I would recommend it both because of the time of day and the length of the trip (7:15) but it is direct.

The fastest is 7:30am that involves 2 changes but only takes (5:43). All the 1 change options take 8+ hours so I personally think the 2 change option is worth if you can save 2+ hours of travel.

This is for Jan 13th - chosen at random in the future - so you'll need to check closer to when you're travelling. Fast train tickets are generally open about 2 months before the date of travel but the schedule is usually available earlier than that for planning.

Have a great trip,
=Tod

Trenitalia still published printed timetables. You can find them here:
https://www.trenitalia.com/it/informazioni/orario_ferroviario.html

I once used the printed timetable to plan a trip around IC services, as those often take quite interesting routes. I would definitely advise a Montorosso - Napels IC if there were one that ran a bit later...

Thanks for all the input! How far ahead should we book the different trains?

If you plan to travel on high speed trains, and would like to save some money by buying ahead of time (up to 6 months), and discounted tickets are sometimes available until a few days or weeks prior to the date of travel. Be aware that there are restrictions on your ability to get refunds or make changes if you change your mind later. If you are not sure about your travel plans, tickets at full BASE price are always available until minutes prior to departure and they rarely sell out, even in the cheapest “standard” class (the higher more expensive classes never or almost never sell out).