Day trips from Florence

6 nights is at least 5 full days. Depending on your interest for art and history you could spend the whole time in Florence and still just scratch the surface, however, on average, most visitors devote 2 full days to the city. So if you are like the average visitor, you could take even 3 day trips.
The Cinque Terre is a long shot. Not far, but train connections make it a long trip. If you just go for the day, I recommend taking in the early morn the FrecciaBianca high speed departing from Firenze Campo di Marte station (Florence second main station) at 7:38 am. That way you can be at the CT in less than 2 hours. All other options (all departing from Firenze Santa Maria Novella, the main station) take considerably longer.
You can see Pisa on the way back to Florence (you must pass by Pisa). 1 or 2 hours are more than sufficient for Pisa and you can do that in the evening. There are almost 60trains a day in each direction from Pisa back to Florence. The last one departs at about 10:30pm.
Other potential day trips are:
Siena
Arezzo (+possibly Cortona)
Lucca and Pistoia (or Lucca and Pisa if you don’t go to the Cinque Terre)
San Gimignano
There are others possibilities but the above are the easiest.

Bologna is also close, 50 minutes by train.

San Gimignano and Siena are best done by bus.

Yes. It absolutely does.
Personally? I think the CT is a bit far for a day trip, and I probably wouldn't do it during the winter. That said, there is one train from Firenze Campo Di Marte station at 7:36 AM in the morning which gets to La Spezia Centrale at 9:12; 1 hr and 36 minutes. This is absolutely the fastest way you can get there.

From there you'll take trains into Monterosso, Vernazza, Corniglia, Manarola and Riomaggiore; the 5 villages of the Cinque Terre. From most northern (Monterosso) to the most southern (Riomaggiore) is only 7 miles so they are just a few minutes apart, and Riomaggiore is only a few minutes from La Spezia.

Coming back, there is one train from La Spezia Centrale at 20:02 that returns to Firenze Campo Di Marte at 21:40. This is the fastest way to get back to Florence.

For your other day trip, I might substitute Siena or Lucca for Pisa. Some folks have been able to combine Pisa and Lucca into one day trip but I'd only do it during months which afford enough daylight.

As others have said, if u are into art and architecture Florence has more than enough to occupy you for a week and more.
For less artsy people ( like me) there are a host of day trip options. But unlike most people in this forum I like to shift bases and see a town at night. Most Italian towns are beautifully lit up at night and a must see IMHO.
You can do two nights of the six in either Siena which is simply beautiful at night and lots to do in the day, or at Lucca which has a atmospheric walled old town and some great low key sites, (walking the walls was a highlight of my trip).
You can visit Pisa as a day trip from Lucca which is a short bus ride away. At Pisa please don't miss the amazing baptistery and the Camposanto cemetery.

Have a great trip!

I would do one day trip in this time to Siena, which is IMHO by far the most interesting and beautiful day trip from Florence. Bus is the best way and it takes about an hour. You can return any time you wish; buses run 2 or 3 an hour.

I hiked the CT 35 years ago before Steves was publicizing it (and has alas made it a tourist trap of sorts) It is still beautiful but the charm of the undiscovered treasure is long gone and it is IMHO too far for a day trip. It deserves a full day to meander the trails (if they are open, there has been a lot of issues with that in recent years due to flooding) Just rushing up there for a quick visit -- too far in a day.

You could take a city bus up to Fiesole - it would even be a possibility to hike back (~5 miles).

Florence is the perfect hub for seeing some amazing places very nearby, cheap and fast by train or bus. Siena, San Gim, Lucca, Pisa, Bologna, Arezzo, Cortona, etc. Florence train station is always crowded but if you buy Trenitalia tickets online before your trip, or at the station a day ahead, it's a breeze. Then you won't panic about standing in line nervous /99about missing the train you wanted. I LOVE the Italian train system!!

Florence is the perfect hub for seeing some amazing places very nearby, cheap and fast by train or bus. Siena, San Gim, Lucca, Pisa, Bologna, Arezzo, Cortona, etc. Florence train station is always crowded but if you buy Trenitalia tickets online before your trip, or at the station a day ahead, it's a breeze. Then you won't panic about standing in line nervous /99about missing the train you wanted. I LOVE the Italian train system!!