Hi there and welcome to the forums!
Ouch, this is really is a lot in a week, and for locations which are REALLY spread out.
First off, you are flying into Rome Leonardo da Vinci, correct? That's in Fiumicino and not Ostia, although Ostia is close. You are going to want to stay in Rome on the evening before your flight home as you do not want to be in Praiano/the Amalfi: it's too far away. Even if flying out of Naples - which is closer to the Amalfi Coast - I wouldn't want to be that far away on same day.
I am assuming you've already purchased your flights?
Assuming that's so, I would rearrange your trip to place Rome at the end of it to be closest to the airport. Take a train directly to Florence on arrival day. There is one or two which depart daily from the airport but if the timing doesn't work, take the express train to central Rome and a fast train from there to Florence.
Praiano: there is no ferry or train service to that one, and the ferries will not be running to all points along the coast in March; it's still winter season there. To get to Praiano will involve a combination of train(s) and SITA bus, or trains/ferry/SITA bus, and it's going to eat up considerable TIME to get there from Florence.
So let's say that there are multiple ways to skin that cat but, personally, I would save Praiano/Amalfi for another trip. While I do understand that you're OK with a "sampling", time = money and especially so when you have limited time to work with! You're trading time you could spend sightseeing for quite a lot dealing with transport, and to locations which don't really line up well to make best use of your few days.
I also think you're cutting Florence and Rome VERY short. You'd barely get your bearings before moving on! If getting a look at some additional places is important to you, you could do a day trip to Siena and/or Lucca if you scratched Praiano/Amalfi and added 2 days to Florence. Add the third day to Rome, which would give you two full days there instead of just one. As March weather could be dicey, that extra day would allow flexibility for a bike tour if one of the two decides to pour rain.
Mar 13: Fly into Rome Fiumicino and train to Florence
Mar 14: Florence
March 15: Day trip
March 16: day trip
March 17 - 19: Rome
March 10: Fly home from Fiumicino
Just a thought?