I'll be in Italy for 8 nights (3 Rome, 2 Naples and 3 Florence) during late September.
1) So my first train ride will be from Rome to Naples. I'm getting 9€ (per person) ticket that too on "frecciarossa"! Isn't it too cheap or there is any catch? I'm 27. I hope I'm eligible to book and board? I know my question can be stupid but I was unable to believe my eyes.
2) My second train journey will be from Naples to Florence. If I look at the direct train i.e Napoli Centrale to Firenze SM Novella then I am getting the best fare as 29€ (the same train FRECCIAROSSA - taking 2.5 hrs.). However on taking a little more pain (I hope it's little?) if I take Intercity from Napoli Centrale to Firenzi Rifredi it'll cost me 9€!!! Though it'll take little more than 5 hrs. (i do not have problem with spending 3 hrs extra in travelling, than paying 20€ extra as we'll be 2 persons! Please let me know if I should reconsider it.) So do you see any problem in it?

Book that 9€ ticket as soon as you can before it goes up - - supplies at the lowest fare are limited. As far as saving the 20€ goes for the longer travel time, it should be fine unless the train is jammed and you have to sit on your luggage in the aisle (though this is not a problem if it is a reservation-mandatory train, like the Frecciarossa).

The only catch, is that these super-discounted tickets have the most restrictions on refunds. If you think you may change your plans and don't want to lose the 9€, buy a higher-priced ticket with less restrictions.

The only catch is the €15 taxi ride from Florence Rifredi to central Florence. Florence SMN is a stroll from most hotels and sights.

The only catch is the €15 taxi ride from Florence Rifredi to central Florence. Florence SMN is a stroll from most hotels and sights>

well you can also take a few euro tram ride from Rifredi station to the center of Florence and I have even walked from Florence center to Rifredi, on the edge of Florence. No need to take a cab and there are also constant shuttle trains you can buy a few euro ticket and hop on. No need to waste money on a cab! simply buy a Rifredi to Santa Maria Novella ticket in say Naples or buy it in Rifredi - zillions of trains running that route that do not require reservations.

Some trains that stop at Florence Rifredi station may also stop before that at Florence Campo d Marte station and if so get off there as this secondary station is just as well situated distance and transit wise from most hotels as is Santa Maria Novella - Campo d Marte is about 1 km or so walk west of the main square if I recall after walking that route a lot.

And if you really want to save from Rome there is a regional train to Florence Campo d Marte that takes only about 75 minutes longer than the high-speed trains and regional trains are dirt cheap (say if you cannot score the 9 euro fare once trying as they can evaporate and actually getting www.trenitalia.com to work can be very problematid (stressing problem) per many many frustrated posters here on Fodors.

Thanks a ton fellas!
This info will definitely help

This is very helpful for us as well. I'm glad someone else had the same question. Thanks for sharing the info