Don't buy anything in advance, there is no reason in your case. Just buy a ticket at the station or on the Trenitalia App when you get there.
The travel BO-RA is aboard regional trains which are like commuter trains or subway trains in America, seats don't sell out, and they are on a first come first served basis (no seats are reserved), if there are seats that are empty, you just pick up one and seat on it, if the train is full and no seats are available (unlikely), you simply stand up.
The portion from Bologna to Florence is likely on a high speed train called FRECCIAROSSA or FRECCIARGENTO. Those seats are reserved (like airplanes) but they very rarely, if ever, sell out, since there are trains every 15-20 min between BO and FI, so you can buy them shortly before you depart.
The only advantage of buying ahead a high speed train ticket is that you can get a discount (discounted fare seats sell out fast and, after they are gone, you can still get a seat but you have to pay the full price). However the distance Bologna - Florence is short, therefore the savings are not huge. In addition if you buy a discounted fare (Economy/SuperEconomy) on a high speed train, if you miss the specific train you book, or if you change your plans, you can't get your money back. Therefore buy your tickets on the spot when you are at the station.
Be aware that the Regional paper tickets bought at the station must be validated (date/time stamped) at the machine before boarding or you get a fine. If you buy via Trenitalia App, no validation is required.