Florence day trips

It’s best to just buy them at the station, either at the ticket window or from a machine.

Or get the app

If it's for short day trips then you'll be riding the Regionale trains which don't require reservations and for which the prices don't change if you book ahead. Just buy as you go.
And do get the App for your smartphone - easy to research and book your rides electronically, plus it will remove the need to validate a paper ticket at the station.

If your not set on taking a train I can suggest a wonderful day trip with Walk About Florence. We took the Best of Tuscany which can include lunch at an organic winery, including a tour of the winery. You can look on their website for the prices and itinerary. I highly recommend this company, the bus was comfortable and the guide very knowledgeable.

Buy when you are there. No discounts for pre-purchased tickets on Regional trains.
The app is easiest.
https://www.trenitalia.com/en/purchase/mobile_ticketing.html

Got this sales ad from Italotreno yesterday. I don't know if it would apply to any of your trips.
"Leave with discounts from -30% to -40%.
Buy a ticket by 7pm on 28/03 using the promo code LUCE and travel from 31 March in Smart and Prima.
Book your ticket now to travel until 21st July!"

I registered for Italo a few weeks ago and I get emails with discount codes almost weekly. I don't have enough experience with it to know if they are cyclical or frequent.

Thanks everyone for the advice. I downloaded the Trenitalia app and will buy the tickets while in Italy. It looks like I can't sign in but I can purchase tickets as a guest. Is my phone all I need to board the trains? I know in Italy you sometimes have to check in with the ticket validation thing when entering the train platform.; I assume you don't have to do that if you buy the tickets through the app, correct?

Correct.

In case you'll ever feel the need to have an open ticket, time-stamping it before boarding should take around 2 seconds. More if you think the "clonk!" noise by the old yellow machines to be Particularly satisfying.

Strange you can't register on the app. Did you download the app from the English app-store?

Register on the Trenitalia site
You will then have an account you can log in to on app

"I know in Italy you sometimes have to check in with the ticket validation thing when entering the train platform.; I assume you don't have to do that if you buy the tickets through the app, correct?"

The short answer is no. Your tickets will be sent to you electronically as an attachment to a confirmatory email you receive after you complete the booking. Though you don't have to worry about validating as you do with paper tickets, you'll need to have the tickets displayed on your phone in order to gain access to the platforms at the larger train stations ... like SMN. In our case the guards barely looked at ours but just waved us thru when I held up my phone.. Likewise you'll need to have ready access to the tickets if one of the checkers inspects once you're on the train - no big deal but you'll want to have your battery charged up just in case.

Thanks Robert, good advice.