We are a family of 9 arriving in July and trying to find the easiest way to get to Florence from FCO and back to rome for a couple of days.
There are a couple of direct trains every day if the timing works out, otherwise take a local train to one of the Rome stations and change to a train to Florence. With 9 people, are they all adults? There are discounts for children depending on their ages. Schedule and prices at www.trenitalia.com. Use Fiumicino Aeroporto as the star point and Firenze S.M. Novella as the destination.
You don't provide specifics about arrival in Rome or the composition of the "family" so I'll make a few assumptions.
First, I'll assume that not all members of the group are of the same age group. If there are young (or people of advanced age) in the group then maintaining group integrity while traveling may be important. As the high-speed trains in Italy are ticketed with reserved seats, the only way to maintain group integrity (all members in the same carriage) when traveling between two big tourist destinations may be to purchase tickets well in advance. However, trying to time a rail departure from Rome on the day of arrival by air can present some real challenges.
Second, I'll assume you are arriving in the morning or early afternoon on an international flight - possibly from North America. If so, you may be checking luggage, which in a group of nine, raises the odds of a luggage-related delay. Bottom line, there are numerous ways your departure from FCO can be delayed so purchasing advance train tickets to Florence may not be advisable.
If you are indeed arriving Rome from an intercontinental flight, my sincere recommendation would be to purchase your Rome to Florence tickets for the day after arrival and spend the arrival night in Rome recuperating from jet lag.
I would book a van service with such a large group and luggage. Will you be renting cars at your destination? Look at TripAdvisor for names of these transportation services.
What Sam said. Also, like many, I greatly prefer going on to my furthest (Italian) destination on the same day as arrival at FCO. That first day is going to be a blur, anyway, with jet lag. Might as well use that blur-time to travel, and save having to do a city and hotel change (which would eat up half a day on your next day.) It's quite easy to take the train from FCO to Termini (or Tiburtina) station and then a quick two-hour fast train to Florence. I'd certainly price out the cost of train tickets vs. a van, though. Finally, although many don't recommend it, I prefer to take my chances and go ahead and pre-purchase tickets from Termini to Florence (or Venice, or Lucca), thus saving some 20-30 euros per person vs. buying the tickets the day of arrival. I do give myself approx. 4 hours between expected landing time at FCO and departure time from Termini. I've played this lottery some 4-5 times (at least) and been okay so far....
I've no idea what actual cost for a private transfer would be but can only image that, for that distance and that many people, it would be very, very expensive. I'm seeing some dummy rates out there that are eye-crossing, to say the least. It will also take longer than a fast train.
So, IMHO, train is the way to go. There are two direct "fast" trains from FCO to Firenze S.M. Novella per day - at 11:08 and 15:08- that can have you in Florence in 2 hours and 14 minutes. Other trains involve a change in Roma Termini or Tiburtina stations and take 15 minutes longer or so. Your options can be explored on the Trenitalia website Sam provided.
You can get some nice price breaks by ordering tickets in advance BUT that is also a risk when coming off a flight, should it be cancelled or delayed. It's a lottery that has worked so far for jmauldinuu but a potentially very costly game to play for 9 people should things not go your way. So, the base price you'll pay at the airport for day-of tickets will be higher but also the safest bet. You do not have to buy first-class tickets: 2nd-class carriages are just fine.
Editing to add: you will be able to order advance tickets BACK to Rome with little risk so will likely realize a nice price break on that end. For that journey, you might also explore prices on Italo. It's a private rail service that serves a lot of Italian cities (but not Fiumicino) and has competitive rates. The stations you want are Firenze SMN and (most likely) Roma Termini and, again, you don't need higher-tier carriages/services: go for the lowest price.
One caveat with those lower-priced tickets on either Trenitalia or Italo: they are usually non-refundable and/or non-changeable so you need to commit to the specific train you're purchasing for. Also, tickets for July are not available yet but should be sometime soon.