I love Italy. Only been twice, but a few suggestions.
known for gelato, you can look online and find some gelato places to go to, my favorite was "cantina de gelato" which was across the river from the Uffizi; this has strange hours tho, closes by 6pm. Best gelato I've had in Italy tho.
Food - for a quick meal I liked Mamma Gina's; to me this reminded me of homestyle italian meals. This is near the ponte vecchio (as is cantina de gelato).
Il Borro - more of a "fancy" establishment (I still went in my normal tourist clothes and brought my three young daughters). I call it fancy because the meal seemed less "home style" and more "haute cuisine". The Chianina beef there was amazing.
Museo de Galileo - neat science based museum, also has the panerai jupiterium there making it a must stop for Paneristi (IMHO). Also attached is a small museum of musical instruments; which is also worth a run through just to ogle the multiple stradivari they have on display.
Favorite store (other than Panerai) would be Alberto Cozzi (Firenze paper products) with a large selection and "Fantasie Florentine" near Mamma Gina restaurant with a very limited selection but where the grandmother clerk still writes your receipt by hand on scrap paper.
Also recommend stopping by the Santa Maria Novella Farmacia - great products with history, nice liquors, and a good place to stop for some tea (they have a small tea room in the back).
Remember to book tickets to the Uffizi prior to arrival and go at the set time.
Skip the gelato here, touristy and terrible (better in the states)(I'm a gelatisti too).
I liked Trattoria da Fiori - they have a prix fix lunch for 50 euro which was amazing and one of the best meals I've had in Italy (50 euro included everything and wine/coffee).
In Murano I picked up some glasses (shipped to me) from a store on the corner of Campiello della Pescieria. Family store, husband and father make all the items, store is tended to by the wife. Very nice people.
There is a restaurant near there (Restaurant - Alla Vecchia Pescheria) which was quite good for a stop to bite and eat.
In Burano (known for lace making) the store I like the most was "Artmoda Barbara". Hand made lace, good quality and pricing. "La Perla" a few doors down is much more fancy, but much more expensive.
Not much for me to say here.
Try to book a tour of the colliseum before getting there so you don't have to wait in line.
Hang out at the piazza's eating gelato, drinking espresso and just soaking in the scene.
You can follow my instagram (akanem71) and I may post some pictures of the places above there so you will have a visual reference.