Cinque Terre to Florence

You'll most likely be able to purchase them when you get there. But most of the good connections between the CT and Florence involve an Inter City (IC) train with reservations, and in theory those could sell out. If you look at tickets for say tomorrow, you still see availability on all trains, though - which to me means there's a pretty good chance you'd be fine booking them last minute.

If you find a solution with just regional trains (and without two changes), those trains cannot sell out so you can absolutely buy those tickets at the last minute for sure.

Use the ticket machines at the train stations in Italy (or wait for an agent) to buy tickets, or use your smart phone: install the Trenit app to buy Trenitalia train tickets and show them on your phone instead of printing, if you wish. Ticket machines take chip and PIN credit cards or cash.

Thank you so much!

That's a fairly short route so you likely won't save that much by purchasing advance tickets. Depending on which departure you choose, you'll be dealing with a combination of Freccia, Intercity and Regionale trains. Only the Freccia and Intercity offer discounts for advance purchase, but keep in mind that tickets are specific to train, date and departure time, so you'd have to be willing to commit to a particular departure. I'd just purchase tickets locally, perhaps the day prior to travel. Which of the five towns are you staying in?

One important point to mention is that if you buy Regionale tickets locally, you MUST validate them prior to boarding the train on the day of travel, or risk hefty fines!

You can check all the rail solutions on the Trenitalia website.

Buy at the station the day you go. I've never pre-purchased tickets in Italy in my life and especially I wouldn't do it from there to Florence where most trains are regional trains (no discounts). Seats are always available.

Andrew's idea of "good connections," meaning those with one change of trains instead of two, depends partly on your departure town. InterCity trains stop at Monterosso but not at the other four Cinque Terre towns. So from Vernazza, etc., your options with only one connection are fewer and are all by regional train. I would buy tickets in the station on the day of travel.

Don't accidentally get onto an IC with a regional-train ticket, since the prices are different and you wouldn't have the necessary seat assignment.

While Trenitalia.com sells the tickets, the Deutsche Bahn link at Looking Up Train Schedules and Routes Online is also handy for all of Europe.