Transportation from Florence airport to downtown

A taxi would be a lot cheaper -fixed rate something like 28 euros? Maybe a bit more for 5
Go to the official taxi rank in front of terminals
No tip needed
( they will call for a taxi van if one is not there)

Tram to station works too

There are several luggage storage locations around the station
Google will find them for you
You can have taxi drop you near one

I take it this is assuming there is no way to store your luggage at your lodging before getting in?

If it were me I would definitely opt for the airport tram to downtown, and not just because of the price.

Even if you can't store bag, or don't feel like church sightseeing with jet lag, you'd still end up taking the walk from train station to Oltrarno. It's a pretty good workout, schlepping bags, possibly uphill depending on where your lodging is. However, since the other option is just to sit around and wait to get in the room, I vote for the walk whether or not you can store luggage, if you think the walk is doable. It totally depends on your luggage! There are a lot of Florentine-style pavings in the center, which could be difficult to roll a suitcase over. I walked through Brno for 2 hours with my bag, because there were (apparently) no luggage storage options, and it was a royal pain.

However, given that limitation, it's a good way to get revived after a flight, and you'd get up close and personal with the Arno, and see the scene. You could even make a detour and go through Piazza Signoria etc. if the luggage again isn't too much.
All of that would fill in the time nicely, as long as you're up to it. If tired at that point, you can stop for ice cream. You will have earned it.

if you have lots of time to wait, I'd go for no. 2.

In that case I would take the T2 light rail tramway to anywhere downtown (the T2 goes all the way to Piazza San Marco now), for less than 2€ each (but large suitcases may have to pay the fare too), drop the bags, visit something, then pick up the bags and walk to your apartment. Just be aware that there are a lot of cobblestones in the city center, so rolling a suitcase may be a little problematic.

There are numerous places that offer luggage storage, you can google it using the Google Maps and find plenty anywhere in the city center.

If you don't think you have a lot of time (in case of airplane or baggage claim delays) and you want to go straight to the apartment the taxi from the airport has a fixed rate (several taxicabs are minivans that can transport 5 people). Since you arrive on a holiday (December 26 is St. Stephen's day, an observed holiday in Italy) the fixed rate is 30€, however they add some supplemental fees for luggage (I think 1€ or so per bag, and also for extra people above 4, maybe another euro or so), but even if you have several bags, you shouldn't pay more than 40€ or so, certainly nowhere near the 100€ you've been quoted by the limo service you may have contacted.

Here's what I would do!

I would message the apartment host and find out if you could possibly check in earlier, and if not, could you leave your luggage in the apartment. Sometimes they will not know the answer to this until that morning, but it doesn't hurt to ask a few days before --- we have rarely been disappointed. They are usually cleaning the apartment. If this works out, then take a taxi to the apartment --- it's really not fun to pull roller bags (if that's what you have) through the streets and I always feel embarrassed about the noise it makes. If you have back packs only, then, sure, walk from the train station.

If you can check in early or leave your bags, then you are free to do anything you want to. If not, how about taking a taxi with your bags to a casual lunchtime restaurant --- I've seen many people do this, including Italians, and have done it myself a few times. If you tell me where in the Oltrarno your apartment is, I (or somebody here) might be able to recommend some places. Eating lunch at, say, 1:00 or 1:30pm might take you nicely to a slightly early check in time like 3:00pm and the host might be able to say yes to that.

I think Roberto's suggestion is what I would do. There are luggage handling places within a block of the T2 tram stop. I would reserve that online ahead of time since they do fill up and it'll save you wandering around that block finding a place that has space.

Thank you everyone!

All of your suggestions are good options which will keep us totally flexible. If our flight is on time, then we will take tram to downtown, store luggage, walk around; if flight is delayed then we will just taxi to our hotel. BTW, the taxi service I was looking at is 110 euro not, 100, my bad. Rome is a lot cheaper...

It’s not that Rome is necessarily cheaper (it’s not), it is that you called a limo transfer service that is out of their mind.

Don’t book anything. When you exit the airport, turn to your right and walk to the taxi stand. There will be dozens of taxicabs lined up. The flat fare imposed by the city is 28€ on weekdays and 30€ for Sundays and holidays. There may be a supplement of 1€ for each bag.

With 110€ you can travel separately in 3 taxicabs if you want, and still save money.