Pistoia and Lucca are very reachable by train.

http://www.flickr.com/photos/mksfca/5941565648/in/set-72157627066754871

Take the bus up to Fiesole. This is not a day trip as it's only a few minutes outside Florence but the views of Florence on the bus are great. There's an archeological museum and amphitheater where you can spend some time.

You can take fast trains to Bologna and be there in less than an hour.

I would not try Milan or Venice---too far.
But, I would include Siena, Lucca/Pisa, Arezzo, and Bloogna.

>>>We plan to take several daytrips: to Venice, Siena and Milan. We will be travelling by train for those trips.<<<

You want to take the bus from Florence to Siena, not the train. The bus drops you at the historic center. The train station is more on the outskirts. The rapid bus between Siena/Florence is also fast than the train.

Thank you for your responses. Are Bologna, Lucca/Pisa and Arezzo worth the time spent? I have a couple of opinions on each: no one seemed very enthusiastic. I love Florence and am convinced that the other cities are worth seeing (even though it means 4 hours on the train) but we don't want to go anywhere just to say we've been.

Whom did you ask? IMHO, the answer is yes.
Do not waste Venice on a hurried day trip. You need 3 days minimum.

Skip Venice and Milan.

San Gimignano (sp?) is an easy day trip from Florence. It could be done in a half-day, even.

My daughter liked Bologna, but I didn't get there.

Your trip sounds wonderful!

There is a new high speed line so travel time to Milan can be reduced now to 1h 45min - short enough for a day trip. Even Rome is within a day trip scope; the line to Venice is a bit slower. But we are talking about a lot of distance so, even if the travel time is relatively short, train tickets are expensive unless you manage to buy a special offer.

Is it at all possible to take a night or two from Florence and spend it in Venice? That would be a given for me and I would skip Milan.
Luca would be a great day trip and even Rome.

The train prices are reasonable if you buy them ahead of time online (few weeks ahead of time) from Rome to Florence. The travel time is about 100 minutes. But I don't know it is safe to buy these tickets online from USA. Did anyone buy tickets on Trenitalia online from USA? What is the process to use them once you are there?

One of our most magical days was a day trip from Florence to Pisa. As much or as little as you want to do, once you're there ... but, for me, it was breath taking. Our pictures don't do it justice. Sadly, it rained the whole day that we went to Siena, from Florence ... but, still very memorable. Both Pisa and Siena are easy day trips, and very inexpensive on public transportation.

As someone said, Bologna is only 45 min. By train.
We stayed a week in Bologna in April...interesting city, well worth visiting.

We just returned from a week in Italy that included Florence, Pisa, and Bologna, as well as San Gimignano (trip report forthcoming). Bologna surprised us with how interesting it was; we followed the "Bologna Welcome: 3 Hours in the Heart of Our City" self guided walking tour and really enjoyed our day.

http://www.bolognawelcome.com/en/places-to-see/suggested-itineraries/params/Percorsi_8/ref/Three%20hours%20in%20the%20heart%20of%20our%20city

I have stayed in Florence and done quite few day trips on the regional trains. Definitely Arezzo, Siena and San Gimignano. For San Gimignano the train stop is Poggibonsi, the bus stop about five minutes walk away and the bus takes you up the hill to the town. You can return to Florence by bus also which was a nice trip. For Siena I took the train and the bus from the station is just a short trip also into the town. I would highly recommend Lucca especially if you like opera as you can visit Puccini's house there. Then you could go on to Via Reggio which I enjoyed very much on a saturday as there was a very good market in the centre of town. I then walked a short distance to the seaside and strolled the promenade. It was a great day out. In these smaller towns you will often find family run cafes where they make their own fresh pasta, it is far superior to the tourist cafes of Florence. I would not try Milan, it needs more time and Venice is just too far from Florence, it would be best imo to give three days for Venice and do a two day stop at Lake Garda or Verona.

I loved San Gimignano and would definitely recommend a day trip.

If you like to walk, we walked from the the Duomo to San Miniato al Monte, a nearby church

http://www.museumsinflorence.com/musei/san_miniato.html

We loved both the church and the views back to the city centre. It was a fairly steep walk so the other option is to use a taxi for the trip to the church and then stroll back into town and take advantage of the spectacular views.

I love that you are spending 9 days in Florence, it's one of my favourite cities, especially later in the day when the tour groups have all gone.

Florence to Venice is 2:05, non-stop, via high speed train. The earliest of those is 08:30, arriving at 10:35. The latest return of the same type is 19:25, arriving 21:30.

So, it is possible, and you would have a little less than 9 hours on the ground in Venice. These direct trains can get pricey, more so if not booked early, so that may be a factor.

An overnight train is not really an option because the only direct one leaves Florence at 01:47, arriving in Venice at 05:20.

You asked if Lucca is worth a day trip - I would give it a decided "thumbs up" and personally rate it much more highly than Pisa.

Arezzo, Siena, Bologna, all a strong "yes"

Milan - I did not find it that interesting, especially if you are deciding between a number of other long day trips.

For smaller places - San Gimginano, Montepulciano, and Volterra are all great stops, but far better options by bus than train.

One place not mentioned that is one of my favourites in Italy; Verona, clocks in at just 90 minutes on the high speed train. The earliest outbound is 09:30, arriving at 11:00, however. The latest 90 minute return train is 18:50, back in Florence by 20:20