Cinque Terra?

Book a room, drop off luggage (even if you have to ask them to hold luggage before the room is available), use trains and ferries to travel between villages.

The correct spelling is Terre (plural).
Every village has a train station. Once you alight the train you can walk your luggage to the hotel or accommodations of your choice. After you drop your luggage you can take the train from village to village. Monterosso is flatter, and I think there might be taxis that can drive you at least to Fegina, the new area of Monterosso. The other villages are pedestrians only and tend to be steep, so the walk from the station would be uphill.

Since you plan to just stay one night just book a hotel in one of the villages while leaving all your luggage at your hotel in Florence. I am assuming that you are planning to return back to Florence.

I would recommend staying at Hotel La Spiaggia in Monterosso al Mare and leaving your luggage there. It is an easy walk only a couple of blocks from the Monterosso train station and rooms with a balcony have a lovely view of the sea. For sightseeing you can take the local train to the 5 villages.

smn,

Is there any possibility you could spend two nights in the Cinque Terre? A one night stay means you'll basically only have the afternoon of arrival day to have a look at the five towns and while that might be possible, it probably won't be the most pleasant experience. What time of year will this trip be taking place?

I would also suggest staying in Monterosso as it's the largest of the five towns and has the greatest number of hotels, restaurants and other tourist amenities, and also the nicest beaches. Many of the hotels have elevators and provide a hearty breakfast. You can use a hotel in either the new town or old town. If you arrive early, the hotel can store your luggage until check-in time. If staying in the old town, you might want to use a Taxi from the station when hauling luggage.

There are definitely "trains that take you" and you can use the Trenitalia website to view all the solutions. From Florence, the trains are often a combination of Freccia / Intercity / Regionale, and at least one change is usually required. If you're buying rail tickets locally, be sure you're familiar with the "rules" pertaining to public transportation in Italy or you could face hefty fines. If you need more information on that, post another note here.

The quickest way to get to one of the five villages is the train from Firenze Campo di Mare at 7:28, changing in La Spezia to the local train:

https://www.e656.net/orario/treno/21182.html [see the villages plus Levanto]

Good train to take if there is a big cruise ship is in port; when being on trains between 10 and 5 best avoided.

How do we transport the luggage from town-to-town? We are "older" and
can certainly handle the walk, but not the walk with luggage

smnjdanty, this is confusing. You definitely will not "walk" with your luggage from village to village, and parts of the "easiest" route between villages have not been totally open for years. If you are expecting to do that, even without luggage (there is NO way you should try and haul your bags on the trails), you need to do some more research about the region.

http://www.parconazionale5terre.it/Eindex.php

What I refer to as the "easiest" route is not easy AT ALL for visitors unused to very steep trails/lots of steps, even when all parts of it were open.

The best thing you can do is choose an accommodation in one of the 5 villages (Monterosso was our choice, and we have stayed at the aforementioned La Spiaggia) and go there immediately from Florence. If you get there during check-in hours, then check in and drop your luggage. If not, ask your accommodation to watch your bags whilst you go off and sightsee. Understand that accommodations with normal hotel-type procedures are rare in MUCH of the CT so just dropping your bags might not be an option.

Yes, you can take trains between the villages. You just need to understand how those trains work, and be prepared to validate tickets for each ride, unless using a pass, so you avoid potential fines.

One thing you haven't mentioned? What time of year will you be traveling? Not all accommodations, such as La Spiaggia, are open during the winter months.