Spending a day in Florence

It would be a very full day if you tried seeing all of those sites, but it's doable since Florence is very easy to get around. I would recommend getting tickets for the Accademia and Uffizi online at the official site (http://www.uffizi.com/accademia-gallery-florence.asp) since you don't want to waste your time standing in line.

You could make an early morning tour of Accademia, then walk south a few blocks to the Duomo. The Duomo and Baptisery are free to walk in, but if you want to climb to the top, you will need tickets in advance.
After seeing the Duomo you can continue walking south to the Uffizi. On your way, stop at the Piazza della Repubblica and Piazza della Signoria.
After the Uffizi, check out the Ponte Vecchio bridge right next door.

I had no time to drop by Piazzale Michelangelo and have a look, but I understand that the free parking has been dismantled about one month ago. Now there are only some 70 metered parking places. But as you are visiting on a Sunday, access to the center and street parking - if you can find it - are free.

While ZTL is not enforced on Sunday, a few bus only lanes still are, so a bit of attention is still necessary.

I agree that you definitely want to get tickets to the Accademia and Uffizi in advance.

Be aware that although it's not too far to walk from Piazzale Michelangiolo to the center (about a mile to the Duomo), the Accademia is about another half mile from there, and in August it is likely to be very hot. Also, on the way back it's uphill (depending on specifically where you park). Bring water, and consider how much wine you have with dinner before walking back.

To answer your question about tickets to the Accademia and Uffizi (absolutely buy them in advance) you have to choose a specific time slot.

This is the official ticketing website:

http://www.b-ticket.com/b-ticket/uffizi/

The Baptistry is near the Duomo, and it's amazing; you can buy tickets at the entrance. You may be thinking of the Boboli Gardens at the Pitti Palace but I'd skip those; you're going to have your hands full enough as it is. I'm unaware of a 'grotto' by Ponte Vecchio but the Loggia dei Lanzi - at Piazza della Signoria - is only a few blocks from the bridge, and has a very nice collection of original sculptures which can be seen for free:

http://florenceforfree.co/2012/07/31/piazza-della-signoria-part-2-the-loggia-dei-lanzi/

You can take a bus back to Piazzale Michelangelo from near the train station, and other points, and you could plan your route to end up at the station to avoid backtracking.

Firenze's Centro Storico is compact. If Palazzo della Signoria is open, it's interesting. The Bargello and Museo di San Marco are also worth seeing, if you have time.

Gelato stops are mandatory and will take a few minutes each.

Thanks everyone for all of your suggestions. I will look into all of them and figure out what we really want t see while there.

Lachera - I will definitely have to look into the parking at Piazzale Michelangelo to see if it is still free or has less spaces that are metered. I wonder if that would also be free to park there on Sundays? (We may try our luck at the free parking within the center - any suggestions on where to go for that? I'm not sure we will know how or where to drive in the big city of Florence.)

Thanks Kathy - I will check out the Loggia dei Lanzi - at Piazza della Signoria for the statues - that's probably what my friend was talking about!

We will definitely look into buying our museum tickets before we go. And of course we will sample the gelato!!

Should you not want to bother with your car, ( although on Sunday, it might be better by car) there is the bus line going from Strada in Chianti right to Florence Centre. It takes around half an hour

http://www.acvbus.it/orari/365.pdf

Does anybody know if the parking at Piazzale Michelangelo is free or metered now? Do you think if we arrive early if we would find a parking spot there? If it is metered now, how do we pay for it and how much does it cost? Any other ideas on parking in Florence on a Sunday?

We have reserved our tickets for the Academia Galleria to see David at 11am, so do not want to run into trouble by not finding a spot to park in once we get there...

Re: Loggia dei Lanzi, there are beautiful original sculptures there. There is a copy of Michelangelo's David in the Piazza Della Signoria (where the original used to be), but why would you go all the way to Firenze to see a copy? Get tickets for the real thing, there are several other things to see at the Accademia as well.

As for free parking, since the water main burst on a street earlier this summer, parking in the city has been even more difficult - perhaps Lachera knows if the work is still going on. I would give up the idea of free parking in Firenze.

Here is a map of parking in Florence. Clicking on the "P" gives you location and hourly fees.
https://www.visitflorence.com/florence-maps/parking-lots-in-florence.html