Rome to Florence by train!

• Trenitalia is the Italian National Rail company, running ~90% of the trains: https://www.trenitalia.com/en.html
• Italiarail s a ticket reseller. It is selling tickets for the trains operated by Trenitalia: https://www.italiarail.com/

What dates are you looking for? Tickets only go on sale 120 days in advance.

Lots of info on Italian trains here: https://www.seat61.com/Italy-trains.htm

What ages are your kids? Note this:

Infants under 4 go free on all trains, no ticket required, just bring them along.
Children means anyone under 14 on national trains (= Frecciarossa, Frecciargento, Frecciabianca, InterCity, ICN), but only under-12s qualify for child fares on regional trains.

Italia Rail is just a US based travel agency that sells Trenitalia tickets. Do you mean Italotreno? They are fine, a little cheaper often, fewer departures, but just as fast.
https://www.italotreno.it/en
What is your travel date? What are the ages of the kids and how many? Trenitalia does have a Bimbi Gratis (kids ride free) program, but it has to be booked in advance and the adults pay the Base fare (full price). The fine print says "limited availability" so if a particular train's quota is met, try another train departure time.

Edit - Yes, paging through a few departures, Bimbi Gratis shows "Sold Out" for this route until September, and even then, 5 SuperEconomy tickets would be cheaper than 2 adults and 3 children with Bimbi Gratis.

I understand Trenitalia's child rates to be under 15 (before 15th birthday) on most trains, but they have used various and potentially confusing descriptions over the years. (Correct that Regional trains are different.) The Bimbi Gratis, Offerta Famiglia, and other discounts all have limited seats available and can sell out. Trenitalia's chart format will display the cheapest offers for the 5 people you enter. In some cases, regular advance-discount rates could be cheaper than group deals. Just don't select a deal that requires a Frecce loyalty card.

Fast and cheap do not go together well when discussing train travel. The fast trains are always more expensive but still relatively cheap in Italy when compare to other countries. If you want to go cheap then use the Regional trains. For example -- a fast train is about an hour, 30 minutes and around 50 euro. A regional train is roughly 4 hrs and about 20 euro. There is variation depending on time of travel, day, etc. If you can grab a discounted Super Economy ticket (lots of restrictions) it could be around 28 pounds. So the cheapest and faster way would be a super economy fare on a fast train.

We are 5 people trying to get from Florence to Rome fast and cheap.

This opening statement seems to be at odds with the title of this post.

Rome to Florence by train!

This is what Seat 61 says about Italiarail. He seems to like it, but the big reason I wanted to share this, is that, if for some reason italiarail works better for you, if you purchase through Seat 61, the $3.50 fee will be refunded. For me personally, I like to use the country specific site when possible, when it isn't my default has been loco2, BUT, I have no experience with Italy, yet. At this point, I've just been studying up.

**www.italiarail.com is easier to use, all in plain English with English-language place names - and it can be cheaper then Trenitalia.com if there are 2 or more of you. It's an online agency which sells Trenitalia tickets in €, £, $ or Au$.
Like Trenitalia, it allows you to select specific seats from a graphic seating plan.
After recent improvements it's a better system than Trenitalia in many ways: It shows a whole day's trains in the search results not just an hour or two, it can book up to 20 people at a time, Trenitalia only books up to 5 people, it highlights the cheapest trains and trains where 1st class is cheaper than 2nd as a free upgrade.
Although www.italiarail.com accesses exactly the same prices as Trenitalia, it can be cheaper for 2 or more people travelling together because it can combine the last remaining tickets at a cheaper price level with tickets at a higher level within the same booking. Trenitalia can't, so will ignore any remaining tickets at the cheaper price level if there aren't enough for all the passengers travelling.
Italiarail charge a small fee around €3.50, this will be refunded for seat61 users if you email them (not me!) at seat61@italiarail.com after booking.**