buying online or at station?

The quickest way to go roundtrip from Rome to Florence is on the Freccia train (fast trains) which require a reservation. There are three levels of cost: base fare (highest), the normal full fare. You can make changes up to the time you travel; since this is full fare, there's no advantage to buying it in advance unless you need a block of seats together. The next level is an economy fare which will save you money if you buy in advance but still gives you some flexibility in making changes up to a certain number of days before travel. The super-saver fare sells out quickly and does not permit changes - if you miss your train you buy a whole new full-price ticket. Sometimes Trenitalia has a special roundtrip in one day fare, but only to select cities. Roninrome.com has very detailed information on this topic.

"Is it possible to buy a ticket for a particular time and use it on an earlier or possibly later train?" No. To amplify Zoe's reply, if you hold a reservation for an Italian train, you cannot use it on other trains, and you will get a heavy fine for boarding an all-reserved train without a reservation for that particular train. If you have a changeable ticket and want to take another train, you have to change it before you board. If you have a non-changeable ticket and miss your train, it just turned into a bookmark or scrap paper. To find fares, look at this guide from Ron In Rome: http://tinyurl.com/c4t693b. On the Trenitalia website, you can see the "base" (full) fare, and if this is acceptable to you, there's no need to buy ahead. All of this applies to the fast trains, which are all-reserved. The Regional trains do not need, or even allow, reservations. Think of them as a "city bus on rails." If you buy a ticket at the station, you can use it on any Regional train, but you MUST validate the ticket before boarding. Again, heavy fines are levied if you don't validate (this prevents it from being re-used).

Thank you for the replies. I will check out Ron in Rome. :)
I must keep reminding myself about validating!

Zoe, You're not the only one. I work on the train system in the UK. I have been writing and reading reminders to validate on this board for a few years. I well understand the consequences of not validating. On my recent trip to Rome and Bologna, with a number of day trips to all sorts, I lost track of the number of "duh" moments either on the trains, with the mad dashes, or walking through the stations, forgetting and then remembering to visit the little machines. I know, I enforce, I forget. duh.

Susan, I've taken hundreds of trains in Italy and I still
have to remind myself to validate the regionale tickets. I am sure it amuses other passengers to see me dash off the train toward the validation machines.