Florence-first time-2full days, 3 nights...day trip??

2 days in Florence gives you just enough time to experience Florence.
If you do a day trip, that will take up your 2nd day; and possibly once you've seen Florence, you will wish you had stayed there both days.
I suppose once you've been in Florence one day, you could decide not to do a day trip on the 2nd day, if you had not made non-cancelable reservations for a day trip.

Ideally you would have a 3rd day to do a day trip to a hill town or the wine country.

Probably the most common day trip from Florence is Siena, take the bus to get there (train is possible but the bus is better). Siena is probably too big to be considered a "hill town", although it has plenty of hills. You're not going to see the wine country in Florence, although you'll be close to it.

There are shuttle/bus tours you can take from Florence to the hill country or the wine country, of course these would take up your 2nd day, leaving you with only one day in Florence.

Kent is right... 2 full days is a minimum for the museums and major sites. You could "go to a town" (it is feasible) but I am not convinced it would enhance your Italian experience (be worth it). Florence (remember it is Firenze on Italian train schedules) is lovely and you will be enchanted. You cannot do everything in 12 days, so do what you can do well.

What month will you be traveling?

Kent is right, if you want to take a day trip while in Florence, you may want to add at least one extra day to Florence.

With only 12 days, I'd concentrate on visiting the big 3, Venice, Florence & Rome.

You could day trip to CT (from Florence) if the weather is nice.

Keep in mind that the day you arrive you will be jet- lagged so that day you won't do much.

Each time you relocate, you will lose about half of that day (checking in & out of hotel, walking to train station, waiting for train,
transit time, walking/bus/taxi to new hotel, getting situated to new location, etc, etc...)

So, 4 nights = 3 full days; 3 nights = 2 full days and so on...

I would add this to the above:
If you decide to give Florence only 1 day,
and if you go anytime from about April to October and/or on a weekend,
and if you want to see the Uffizi and/or the Accademia (The David) on your one day in Florence,
then you'll probably want to make advance reservations for those two. If you don't, the long ticket-buying lines at those two could take up a lot of your one day.

If you were in Florence longer, you could buy the Firenze Card which allows you to skip the ticket-buying lines, but for essentially a one-day visit the price of the card won't justify the cost.

In general, Sundays and Mondays are not ideal for seeing lots of sights efficiently; some sights close early, even before 2pm. And both the Uffizi and the Accademia are closed Mondays.

I would suggest spending 3 days and 4 nights in Florence and then adding another day to see Tuscany wine country. Having lived in Florence, I have to say that even 3 days here will leave you aching for more. No town would be better to see than Florence and it will keep you quite busy!

About an hour and twenty minutes away by train is Arezzo, which may fit your hill town requirements. You can get there by bus as well.

For twelve nights, I would spend 3 in Venice, 2 in Cinque Terre, 4 in Florence (including a daytrip), and 3 in Rome. Or 3 in Florence and 4 in Rome with a daytrip to Orvieto (white wine) from Rome.

I also would add a day to Florence. Our first trip to Italy our daughters were 17 and 19. We visited Rome, Florence, Naples and Venice for 17 days. Our daughters' favorite city was Florence where we spent 3 full days.

Thanks for all the great suggestions & ideas. I had a feeling that was going to be the consensus from everyone. We do realize with the train travel that we we only get 2 FULL days & I know they will be filled with things to do. For the post referencing the teenagers, just curious, what about Florence did they love so much? Want to pass it on to my teenager.

You could also do a day-trip to Cinque Terra. Could go on your own on a train or your hotel can arrange something for you. We did it on our own.

3 nights/2 full days may be a bit short considering that you will probably use all of that in Florence alone. You might need an extra night to entertain that thought.
You could take a bus trip to Greve and back, if you want to see the Chianti.