Hello juice06870,
The car rental site says 10 minutes for the shuttle that connects the car lots to the airport itself.
There is now a tram that connects Florence airport with the train station - it takes about 20 minutes and the trains run all the time - probably every 5 minutes or so during the day. You will need to take the car rental shuttle to get the airport and then the tram into the city. I'd guess 10 minutes for the shuttle, 20 minutes for the tram and with waiting and family travel time on both ends I'd guess about an hour.
Airport shuttle schedule is here: https://www.gestramvia.it/orari/
You could also take a taxi for convenience since you're travelling with four people and luggage it might make sense just for keeping everyone together but the time will probably be about the same depending on traffic. Taxis are a fixed rate of 22e but could easily be worth the convenience. Note: Always take taxis only from organized taxi stands.
Hope that helps,
=Tod
Also I could have missed something but the only Alamo car rental location I saw was in the city - not at the airport.