Train from florence to Varenna

On both of these trips, you’ll have to change trains in Milan. The trip between Milan and Varenna is on a Regionale train and buying a ticket in advance is not necessary. If you can commit to the Florence-Milan and Milan-Rome at specific times you can save money by buying tickets early. But if you miss the train, the tickets are worthless.

For fares and schedules use the TrenItalia website. When entering cities use Italian spelling.

There are direct high speed Freccia Rossa trains (no change) from Florence to Milano Centrale. Then in Milan you need to take the Regionale to Varenna.
Same to go back down to Rome. Regionale from Varenna to Milano Centrale, then high speed Freccia Rossa to Rome.
You can buy a BASE fare (full fare) ticket on the spot at the station just minutes before departure.
If you prepurchase online ahead of time (up to 120days prior) you can save money on the Frecciarossa.
Just be aware that discounted tickets (Economy and Supereconomy are restricted: no refunds or time changes once a specific train is booked. Timetables and prices below. Use Italian names of cities.
www.trenitalia.com

Here’s a good tip - when you get on the train from Milan to Varenna make sure to sit on the west side of your car. That will give you the lake view while on the train!