What would be a good day trip from Florence, by train?
You're spoiled for choice!
1. I agree Siena is well worth visiting, though I'd take the bus from Florence rather than the train.. The bus is faster and stops at Siena's historical centre, whereas Siena's train station is quite some distance from its historical centre: a 20+ minute walk (or bus or taxi).
a. Take the train from Florence (Firenze S.M. Novella station) to Lucca (from 1h 19m) and explore the lovely walled Medieval town (the train station is a couple of minutes' walk from the entrance through the walls).
b. Then take the train from Lucca to Pisa San Rossore station (less than 30 minutes). You can walk from the station to the Tower, Duomo, Baptistry and Camposanto in 10 minutes or so.
c. Walk from there through historical Pisa itself (there's much more to see than just the Tower etc) to end up at Pisa Centrale station, where you can take the train back to Firenze SM Novella (around an hour).
3. A daytrip by train to the Cinqueterre is perfectly feasible and, with an early start, it is possible to visit all 5 CT villages. Take the train from Florence to La Spezia and change there for the frequent trains which run along the coast, most stopping at each village..though do be aware that Corniglia is high above the sea: it's a steep haul upwards and downwards from the station!
4. Other possibilities:
Viareggio (from around 1.5 hours) if you want to see the sea.
Arezzo (from just under an hour): Roman amphitheatre, Medici fortress etc.
Bologna (just 35 min by high-speed train), a fascinating 'foodie' city with some magnificent historical sights. Don't miss the Due Torri and the wonderful Basilica Santo Stefano, a complex of seven ancient churches.
Train times, details and fares in English on the official Italian railway website:
You don't need to buy tickets in advance for any of these journeys except the high-speed trains (fares for those are cheaper in advance and, as these trains don't allow standing, they can sometimes can get full-up). Otherwise, just buy tickets from the station on the day you go (or the day before, to save a bit of time). Ticket machines all have English language options and there's always at least one machine which accepts cash as well as cards.
Enjoy choosing! :-)
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Tak a bus to Lucca. The train station is very far from the city center. Lucca bus depot is not far from the center of Lucca.
THE best thing to do out of Florence is the Tour of Tuscany https://www.walkaboutflorence.com/tours/best-tuscany-tour
I've done it twice and both times were AWESOME. Seriously. Do it.
I have to put in a vote for Siena. It is by far one of the most extraordinary places I've ever been in my life.
Venice is one of the hives your European tourism attracting over 60,000 tourists on a daily basis. While the colorful houses, winding ancient streets, gondolas smoothly gliding on the canal and carnival masks peeking at you at every corner offer a once-in-a-lifetime experience; the crowds can undoubtedly make their unpleasant mark on your holiday idyll.
I vote for Siena too. It's easy to get from Florence to Siena. Train leaves almost every hour from Florence. ItaliaRail's online info is top notch. You get travel distance, travel time and cost and frequently asked questions. Take a look: https://www.italiarail.com/pages/routes/florence-to-siena
If you haven't been to Pisa, I suggest a day trip to Pisa. It takes under an hour by train but you'll need to walk for about 20 minutes from the train station to the Leaning Tower. For comfort and ease of travel, join a half day guided tour from Florence to Pisa. It's inexpensive, usually less than $60 per person. Take a look at this half day Pisa and Leaning Tower tour on Viator.