Siena

Siena is my favorite place in Italy so it is difficult to say what I would do if I had only one day but I think it would look something like this:
1. In the morning I would go to a pasticceria (pastry shop) for a coffee and pastry for breakfast. This is a good place to observe Italians being Italian. Notice that most everyone stands at the counter visiting with a friend. If you happen to be in Siena on a Wednesday visit the street market across from the main bus station.
2. Visit the Duomo and while there enjoy the Piccolomini Library. (With only one day I would not visit the Baptistry or museum.) (An early visit, or a late one, will help you avoid the many, many tour groups.)
3. Then follow one of Rick Steves walking tours to get a sense of the city.
4. End your walk at the Campo.
5. For lunch go to Pizzaria San Martini on Via San Martini about a 5 minute walk from the Campo. Have a traditional sandwich on very thin focaccia bread and a glass of house wine. (Total cost about 5E.) Sit at a table out on the street if weather permits and watch the life of the city go by.
6. After lunch wander back to the Campo and visit City Hall. You will learn a bit of the history of Siena and of its famous horse race - the Palio.
7. Stop into a gelatoria and have an ice cream.
8. Wander the streets aimlessly and enjoy the ambience. If you want to visit a beautiful ceramic shop go to Ripodzioni d'Arte Senese on Via delle Terme, 74.
9. Sit down at an outdoor table at one of the restaurants in the Campo. Have a glass of wine or a Spritz (it will come with snacks) and enjoy people watching. Stay as long as you want.
10. Have dinner at Osteria al Palchetto on Via San Martini for a truly traditional Tuscan dinner.
Enjoy your visit and hope you will visit again when you can stay longer.

You just got some very excellent recommendations! I would take issue with #7 though. Ice cream? Gelato and ice cream are very, very distant cousins from parents who never met each other. Over several years I've really gotten hooked on Trattoria Papei for dinner (the owner recognizes me now). It's on a market square behind City Hall. Outdoor seating, community (not touristy) feel Nice view too. Drinks on the Campo are fine but the food is less than worth it. The snacks that come with your drink will get you to dinner.
Wandering in Siena is easy to do and it is very easy to discover the different neighborhoods (contrade). It's a real city and has been for centuries!

We had been to Siena twice before taking a Toursbyroberto tour ….. take his tour! AMAZING.

Thank you everyone! This is just what I was looking for. I can't wait!