wifi on fast train Rome-Florence

Yes, on the .ItaloTreno trains.

No wifi available on FrecciaBianca.

The two higher levels of Freccia do have wifi.
http://www.trenitalia.com/cms/v/index.jsp?vgnextoid=f5a0d267965cf310VgnVCM1000008916f90aRCRD

Thanks.

In my experience, I have always found free wifi on Frecciarossa trains, but as the password is sent as a SMS, you need an Italian cellular phone number in order to get it.
If memory serves me, the number of your ticket is also the wifi password on Italo, but I used this service only once and I am not really sure about this.

.italo website says, "with either a temporary username and password sent to you in a text message or your italotreno.it username and password. " The latter refers to the "Italo Più Loyalty Programme."

I imagine the staff in the .italo lounges can get you a password before you board, as well. I'm booked on .italo from Roma Tiburtina to Bologna in a couple of months, so I'll find out for sure.

As well as using an italian cellphone number you can also register using a credit card for 1 Euro CENT. Just turn on wifi on your phone, connect to the freccia network and open your web browser. You will then receive instructions on how to connect by sms or credit card.

The wifi on .Italo no longer requires an SMS. The new procedure is to connect to the network, fill in a short form with ticket PNR code and a valid email address. You then have a decently fast connection, much improved over previous coverage, for one hour. If you will be on the train over an hour you click on a link in an email they send to the email you provided in the form. It comes after about 15 minutes and a simple click keeps you on for the duration of the trip.

I haven't recently tried the Freccia version of wifi as my recent trains have been Bianca rather than the two higher levels.

Italo wifi is super easy. Save your ticket PNR.

If you have questions just ask a representative in the customer lounge before you board the train.

Wi-fi was free and easy to use on our three Italo high speed trains fall of 2015, booked in the cheapest class. We absolutely loved Italo. In our case it was the cheaper option each time, and we found a Naples to Rome train that fit our schedule and ended in Termini.

We had no international plan (Verizon, iphones) so kept cell data off the entire trip. As long as you have your PNR code like others stated, it's easy to sign up.