transportation from Florence airport to lodgings

Taxi stand in front of terminal - fixed rate 22.00€, more on holidays and at night:
https://www.aeroporto.firenze.it/en/the-passengers/transport/taxi.html

Unless you are flying into Florence in the early morning /late night, I'd just get on the first cab in the line outside the arrivals.

The fixed fare to downtown is 22 Euro plus € 1 for each suitcase. Ask a confirmation before getting on, but all fixed fares should be written somewhere near the Taxi stands.

Taxi.
Unless things have changed recently the taxi stand is next to the old terminal. When you exit the airport terminal when you arrive, head to the right. The taxi cabs will be about 50 m away.
The taxi ride to the city center is short (~4 miles) but if you arrive in late afternoon/early evening the traffic is not too good in Florence. The earliest arrivals at FLR are after 9:30am so you won’t be affected by the morning commute too much if you arrive in the morning.

Do the taxi's take a credit card, or plan on bringing cash for the taxi?

Taxi's are supposed to accept cards, and starting June 30, 2022 any commercial merchant which does not accept cards will be subject to a fine of Euro 30.
Having said that, some taxi drivers find all kind of excuses not to accept electronic payments ('my POS machine is momentarily broken', 'there is no cell signal here and the POS machine doesn't work', 'there is a minimum of Eur 30 to use the card', and so on). In reality they are just excuses, because Italians like to cheat the taxman. Some cities are more virtuous than others, but in an article last month I read that only 50% of Neapolitan taxi drivers have working POS machines. The city of Florence is the most virtuous. Nearly all merchants accept card payments. Hopefully when the new law comes into effect on June 30, this issue will be over, but I don't know. If I were you I would have some cash with you just in case. The taxi ride from the airport in Florence to the city center is less than 30 euro. In Rome from the airport to the city center it's 50 euro flat fare. Within the city you will rarely pay more than 20 euro on any given ride.

And is that cost for the taxi per person or per ride (there will be 4 in my group)? Thank you!

Taxi fare is per ride for up to 4 passengers.