Train illiterate😵

All the times include at least one additional stop.

I think you mean "one additional change". Most connections seem to have a change of trains in Pisa.

Looking at the Bahn website, I see one (only one) direct connection from Florence to La Spezia. It's an ES from Firenze Campo di Marte at 7:38. Nothing else direct , at least until afternoon. I get the same results from the Trenitalia website.

Not sure what you mean by "the smaller station in Florence". Do you mean "Firenze Campo Di Marte"?
All trains from Firenze Campo Di Marte to La Spezia would involve a change at "Firenze S. M. Novella" (the main station), so why not start there?

Yes, all trains from Florence (Firenze S. M. Novella) to La Speza involve a change (not a stop) at Pisa Centrale. For each leg (Florence to Pisa and Pisa to La Spezia) the train would have other stops as well.
For example:

Firenze S. M. Novella depart 08:28, Pisa Centrale arrive 09:28
Pisa Centrale depart 09:46, La Spezia Centrale arrive 10:44

Note, most of these are Regionale, so there is no price advantage for advance purchase.
All these times are from the Trenitalia site: http://www.trenitalia.com/tcom-en

Especially for a day trip, that Freccia train is your best bet. It will cost twice as much as using Regionali (27 EUR), but it will get you there quickest in the morning. Campo di Marti station is just on the other side of the old city center from the Santa Maria Novella side.
Coming back to S.M. Novella, there is a direct Regionale at 6:54 pm for 13.60 EUR. Or you can wait for the Freccia to C. d. Marti at 8:03

As far as I know, this is the fastest way to the CT:

There is a train from Firenze Campo di Martre directly to La Spezia Centrale (no it doesn't stop at Santa Maria Novella) at 7:38 in the morning. It takes only 1 hr, 33 minutes, and from La Spezia you are only a few minutes by train into the CT.

Coming back, there is a train leaving La Spezia Centrale at 20:03 directly to Firenze Campo di Martre and takes 1 hour, 37 minutes.

There is one train that leaves La Spezia at 18:54 direct to Firenze Santa Maria Novella but it takes 2 hours and 48 minutes so considerably longer than the direct train to Campo di Martre.

Use the Trenitalia website to explore your options.

http://www.trenitalia.com/tcom-en

There is no need to purchase train tickets in advance! Just buy them the day of your journey.
Note: Firenze Campo di Martre station is on the far western side of the city's center: you might want to take a cab to your hotel, or bus (I don't have schedules but someone else might) to get closer to where you're staying.

Thanks! So if it's a regional train, it will stop at both stations in Florence anyway, but the morning freccia train suggested will depart C d Marte and not stop at SM Novella?

Also, if it's not the regional train, we should purchase ahead of time for best price, correct or no?

the morning freccia train suggested will depart C d Marte and not stop
at SM Novella?

Right. It will go directly to La Spezia, no stops.
The 7:38 train is a high-speed FRECCIABIANCA - not a regionale - so you could buy in advance if super-economy or economy tickets are available and you want to save a few euros but you MUST make that train or your tickets are essentially voided. You will have seat assignments on this train.

The 20:03 coming back is also a FRECCIABIANCA and not a regionale so the same applies.

If you go just for the day, I would seriously consider the high speed FrecciaBianca from Campo di Marte, even though an early wake up is required. All other options take about one hour longer, and many require an extra change at Pisa. With the FB you can be at the CT shortly after 9:30am.
Coming back in the evening it may not be as important, but you want to make sure you get there as soon as possible in the mornings.
No need to purchase in advance, you can buy at the station. The trains stop at Firenze C.M. only 2 min, so try to be there on the platform a few minutes prior to train arrival time (7:36 in this case). It's a short taxi ride, so not that expensive. There are buses that go there, but unless you stay near Piazza San Marco, I'd take a taxi.

Actually, that Frecciabianca does stop for 2 minutes at Pisa Centrale and Massa Centro. It is coming from Roma Termini on the way to Genova Piazza Principe.

Actually, that Frecciabianca does stop for 2 minutes at Pisa Centrale
and Massa Centro.

Right you are, Sam. I should have said no CHANGES for the OP versus no stops.

Thanks much! I should have been more specific. I did mean "changes" rather than "stops". However, another question follows...do they announce upcoming stops, or is there an info board or do we just need to know how many stops there are until our stop?

Sometimes the stops are announced, sometimes not. On the Freccia trains they usually are, but one trick I use is to write down the name of the stop befor mine, or take a photo of the Departures poster (yellow) that shows the stops and the expected times of arrival.

Thanks! That's a good tip!

PA systems on trains are a relatively new thing, and sometimes they don't work well, or sometimes people have headphones.

The way I used to do it, and still do now to a great degree is to know the names of the intermediate stops, and their times. I have printed out the trip with the intermediate stations. If I have an app, like the Trenitalia app and signal I can follow in real time, but usually I make a screen snap and follow the stops in the photo app because signal drops at just the wrong moment.

And I look out the window. Every station has the name many times on signs on the platform (in Italian of course, so it is a good idea to know the Italian names for cities).

Always know the name of the stop prior to the one you need and about how long between it and your stop. Be at the door ready a couple of minutes prior to arrival ready to make the next more....

The scheduled arrival time will be written on your tickets: trains can be late, but they can't enter train stations earlier than expected.

Alecica. I believe you already have the info for getting around by train in the CT itself? You made a comment on the tail end of a previous thread on that subject that indicates that you do. :O)

As far as I know I have the correct info. A friend always used to say "you don't know what you don't know".... it is my understanding you purchase a Cinque Terre card once you arrive in La Spezia, or at the first stop in one of the CT towns. This let's you use the trains for a 24 hour period and also includes bus passes, but not the ferries. Is this correct?

it is my understanding you purchase a Cinque Terre card once you
arrive in La Spezia, or at the first stop in one of the CT towns. This
let's you use the trains for a 24 hour period and also includes bus
passes, but not the ferries. Is this correct?

Close but not quite. :O)
The pass is good for the DAY that you use it: it expires at midnight on the day it's validated so no, it doesn't go 24 hours. That aside, it looks like you're just taking a day trip so you will be fine with a 1-day card.

http://www.parconazionale5terre.it/page.php?id=158

No, it doesn't cover the ferries or your train from La Spezia back to Florence.