Danielle,
It would help to know where on the Amalfi Coast you want to go? The route will vary somewhat depending on the destination.
As the first reply mentioned, it might be faster to take the Corse Rapide (Expresss) Bus from Siena to Florence (about 1 hour), and then connect to the high speed trains.
If your destination is Sorrento, travel time from Firenze SM Novella to Napoli Centrale via Freccia train is about 2.5 hours, depending on which train you use. From there you'll walk downstairs to the Garibaldi station and buy tickets for the Circumvesuviana for the trip to Sorrento (about 70 minutes).
If your destination is the town of Amalfi, it will be a somewhat longer trip. One method you could use is to travel via high speed train from Firenze SM Novella to Salerno (about 3H:25M, depending on which train you use). From Salerno you'll have to take Bus or Taxi to Amalfi. Another option would be to travel from Salerno to Vietri sul Mare by local train and than Taxi or Bus from there to Amalfi.
You can book your tickets for the high speed trains in advance, but you MUST be willing to commit to a particular train, as those have compulsory reservations which are specific to train, date and departure time. If you miss your assigned train, your tickets may be worthless (depending on what class of ticket you buy). You can save money if tickets are purchased up to 120 days in advance. However if you'll be travelling in the near future, all the cheap tickets may be sold out and only Base fare available. In that case you can either buy your tickets in advance online, or buy locally when in Italy using either a Kiosk or at a staffed ticket office, as the price will be the same. For travel on Buses or Regionale trains, you MUST validate the tickets prior to boarding or risk hefty fines.
If you want to purchase tickets online, you might try www.capitainetrain.com as I've found it really easy to use. Note that on both the Florence - Naples and Florence - Salerno routes, you can also use the competing Italo Treno. Check the prices both there and at Trenitalia.com and take the best deal (you can buy tickets for both rail networks using Capitaine Train).
Once you've decided on a specific destination, it will be easier to offer more detailed suggestions. Sorrento is probably the easiest place to use for a home base. If you use the Circumvesuviana, be sure to wear Money Belts and watch your luggage closely, as that train has a bit of a reputation for pickpockets and scammers.
