Terri,
As the others have mentioned, you don't "need" to pre-purchase any of the tickets, but if you want to save some money you could buy the Florence - Naples and Naples - Rome tickets in advance as those will be via Freccia high speed trains. However one important point to note is that the tickets will be specific to train, date and departure time, so can ONLY be used on the one train listed on the ticket. If you miss that train for any reason, your tickets are worthless and you'll have to buy more at full price (Base Fare).
A few notes on the other routes you mentioned.....
- Milan to Varenna - that's a Regionale so not necessary to buy in advance as there are never any discounts. If using locally purchased Regionale tickets, these MUST be validated prior to boarding the train on the day of travel, or you'll risk hefty fines which will be collected on the spot! Are you taking that train immediately after arriving at MXP? If that's the case, you'll also be using the Malpensa Express from the airport to Milano Centrale.
- Varenna to Cinque Terre - the first segment from Varenna to Milano Centrale will be Regionale. You might want to buy your ticket for that segment prior to going to Varenna as the station isn't staffed (although it's also possible to buy tickets in Varenna at the helpful Tivano Tours office). From Milano Centrale to the Cinque Terre may be Regionale or one of the express trains, depending on which trains you decide to use. The express trains have compulsory seat reservations.
- Cinque Terre to Florence - that trip will probably require at least one change in La Spezia or Pisa (or both). The trains will typically be a combination of Regionale or Express, or just Regionale. On this route it's just as easy to buy tickets locally as you won't save much with pre-purchased tickets for the Express trains.
- Florence to Naples - this trip will be via high speed Freccia train, so you could certainly buy your tickets now. You could also use the competing Italo Treno on that route. Note that to connect with the Circumvesuviana, you'll need to go downstairs to the Garibaldi Station in Naples (watch your luggage on that train as it has a bit of a "reputation"). I'm not sure the Campania Express will be operating at that time of year, but that would be a better option than the Circumvesuviana.
- Naples to Rome - again, take the high speed Freccia or Italo trains on this route, using advance tickets if you wish.
Which of the five Cinque Terre towns are you planning to stay in? You'll need to know that when you buy tickets, as the "Cinque Terre" is not a rail destination as such. One point to mention is that some of the tourist businesses start to shut down for the season at that time of the year, so booking a hotel in advance would be a really good idea.
You can buy tickets for both Trenitalia and Italo Treno on the www.trainline.eu website. Once registered on the site, it's very user friendly.