Florence in evenings

Palazzo Vecchio:
http://museicivicifiorentini.comune.fi.it/en/palazzovecchio/informazioni.html

Some of piazzas are actually as much or more fun during the evening than during the day: Piazzale Michelangelo, Piazza del Duomo, Piazza della Signoria, Piazza della Repubblica.

Eat:
Mercato Centrale:
http://www.mercatocentrale.it/en/

Ciro and Sons:
http://www.ciroandsons.com/eng-index.php

Francesco Vini:
https://www.francescovini.com

Personally, I like Florence better at night. Spent about a half-hour on once cold March night examining up close the detail on the Duomo and the Baptistery, something I wouldn't have been able to do during the day with the throngs of tourists almost at any time of the year.

Kathy, I think Mercato Centrale might not be open at night unless you go upstairs to Eataly, that Mario Batali-inspired crap show for tourists. Visiting the downstairs area during the day, grabbing a beef sandwich at Da Nerbone, which has been open since the 1870's, is the thing:

https://www.walkaboutflorence.com/what-to-do/da-nerbone

Kathy recommended Ciro's, which is bar none my #1 choice in Florence. Also, Toto serves a different version of Bistec alla Florentine, sliced thin and laid on a bed of arugula, dressed with high-powered balsamic. Excellent.

http://www.ristorantetoto.it/en/

Finally, Caffe Gilli is the most famous pastry shop in Florence. Definitely worth a stop if you're not having gelato, which is everywhere.

https://www.facebook.com/CaffeGilli/

Kathy, I think Mercato Centrale might not be open at night unless you
go upstairs to Eataly,

Yep, the downstairs market is awesome but not open all day. I didn't recall that the food hall upstairs looked all THAT touristy. I also believe that Eataly is just one shop on that floor but then, my brain isn't getting any younger. HA. Anyway, it's open until midnight, and you can see a map of the layout if you scroll down to the bottom of this page:

http://www.mercatocentrale.it/en/mercato-centrale-florence/

"Not only can you do your normal shopping, but at Mercato Centrale Firenze, you can also find 500 places to sit and enjoy wholesome fare guaranteed: all operators are committed to a high standard of quality and all dishes prepared inside the Mercato are made using only the raw materials on sale inside the Mercato itself. All this makes Mercato Centrale Firenze the perfect place for any time of day, from 10 am until midnight."

To determine which museums, etc. will be open on the evenings you're in Florence, it's best to refer to guidebooks or pick up a list at the TI office upon arrival. Outdoor sights such Ponte Vecchio and Piazzale Michelangelo are of course open 24/7.

There are dozens of good restaurants scattered throughout central Florence, so almost any area will have a few. Most are away from main squares, with Piazza Santo Spirito area maybe being an exception. The second floor food gallery in Mercato Centrale has become very popular with Italians, and you'll now find a good mix of locals and tourists in the evening (seems much more touristy at lunch). It has a large variety of different food vendors, with Eataly having only a small retail spot. Many enjoy it, but personally I'm not a big fan - very hectic, a little pricy.

There are a couple of pubs on Piazza del Duomo. The Aussie pub tends to attract a younger crowd while the other one attracts a more subdued crowd. There's an Irish pub (Fiddler's Elbow) on Piazza Santa Maria Novella. There's another Irish pub and a craft beer pub on Via Faenza. There are many others (keep an eye out as you walk around), but these will get you started.

One of the major museums has a collection of "extra" statues outdoors, someone else will have to jump in w the name. Lovely chance for some photos, and its free.

a collection of "extra" statues outdoors,

You're probably thinking of Loggia dei Lanzi/Loggia della Signoria: the 14th-century porch on Piazza della Signoria, attached to one end of the Uffizi. Some of the collection was actually assembled or commissioned intentionally for that space per an underlying, politically charged theme. It is free and well worth a visit!

Unless that has changed, they did close access to it at some point in the evening - there were guards in attendance when it was open - but I can't find anything regarding hours.