Travel between Rome and Florence for only 12 Euros: now that's an eye catcher! But it's absolutely true: for less than the cost of a taxi ride in central Rome, you can travel all the way from Rome to Florence by Trenitalia Intercity (IC) train. In fact, the 12-Euro advance purchase "MINI" fare in second class is even lower than the undiscountable (except for children) fare for the unreserved "regionale" train (17.15 Euros in second class). The 12-Euro "MINI" fare on the IC train represents a 60% discount over the regular, undiscounted fare of 30 Euros in second class.
Most of the IC trains between Rome and Florence pull into the Firenze Rifredi Station (where you must connect to a "regionale" train for the short hop to the main SMN Station). However, there is one IC train in each direction which departs from and arrives at Firenze SMN: it's the 18:20 IC from Rome Termini (arrives Firenze SMN at 20:58), and the 5:50 a.m. IC from Firenze SMN (arrives Roma Termini at 8:30). This particular train does not run seven days per week in each direction, so look up the schedules for your expected travel date.
Schedules for the IC trains not convenient? No problem: find a "MINI" fare on the ultrafast Frecciarossa or Frecciargento train between Rome and Florence for as little as 19 Euros (second class), and you'll arrive (nonstop) in merely 95 minutes using the new high-speed rail line. But, to get this low fare, you'll likely need to book well in advance, because the "MINI" fares are capacity-controlled, and consequently the lowest fares sell out quickly, especially on trains departing during high-demand timeframes during the day. Frecciarossa/Frecciargento trains between Rome and Florence run every 30 minutes or less throughout the day.
Finally, there is always the tried-and-true unreserved "regionale" train, which costs (only) 17.15 Euros at any time, but takes four hours to reach its final destination! Unfortunately, there are fewer "regionale" trains these days between Rome and Florence.
If travelling with children under 12, also consider the "Familia" fare or (if not available) simply the 50% discount for children between the ages of 4 and 12. (There is no special "MINI" fare for children, and children pay the same "MINI" fare as adults. Naturally, if the "MINI" discount is less than 50%, the child's fare should be selected since it's more advantageous). Also note that the "Familia" fare will not always result in a lower total fare than purchasing "MINI" fares for both adults and children (or purchasing a "MINI" fare for the adult and a separate child's fare).
Children under 4 always travel free without a guaranteed seat: you do NOT count them when booking seats.
Finally, don't forget to read the rules and strict restrictions of the "MINI" fare before selecting it ... or you might be sorry!
(The author of this post has no relationship with Trenitalia).