Should we rent a car or take the train from: 1. Florence to Rome
2. Rome to Amalfi Coast
Answer to point one is absolutely train. For point two I would also go by train but it depends on your plans when you get to Amalfi Coast.
sue, I'd also recommend travel by train on both routes. Not only will it be quicker, but no worries about tolls, expensive fuel, IDP's, ZTL or traffic tickets, CDW costs, parking or vandalism. Also, driving on the Amalfi coast can be "challenging". The trip from Florence to Rome is only about 1H:35M via EuroStar Italia/Alta Velocita and a very pleasant trip. The trip to Naples via ES/AV is only about two hours, and then whatever time is required to reach your destination on the Amalfi coast. It would help to know where you'll be going on the Amalfi Coast, as the transportation varies? One of the easiest ways to reach Positano is by Circumvesuviana train to Sorrento and then Bus. Happy travels!
From what I know about the Amalfi coast road, it is scenic but treacherous. You'd be better off taking a bus so you can give your undivided attention to the scenery.
Sue - Unless you tell us you have a specific reason for a car, train it is. The Amalfi will need some buses too, but driving in general is not a great option for most tourists.
Sue all the major sites you need to see on the Amalfi Coast, eg Amalfi, Ravello, Positano can be reached by SITA bus. Much more enjoyable and relaxing than driving this strech of coastal road. You can take a boat from Amalfi to Positano and over to the Isle of Capri.
You can also visit Pompei by taking the bus to Salerno and take a train to Pomepei