Perche

There are several websites. One is Home Food, volunteers supported by the Ministry of Agriculture. They are an association of about 500 homes where you can cook and eat with a great cook for about 50 euros per person, food, wine, all included. No tipping.
http://www.nytimes.com/2010/04/11/tr...ugal.html?_r=0

http://www.chicagotribune.com/lifest...ry.html#page=1

http://www.classetouriste.be/home-food-italy/

Where it says "Where," just type in the city.
http://www.homefood.it/en/

http://www.newsday.com/travel/eating...omes-1.2591509

http://www.newsday.com/travel/eating...ly-1.2593021#7

Bon Appetour is a good one. You can eat in homes in hundreds of cities in Italy with them. Look at the pictures and read this experience for Rome to get an idea of Bon Appetour, then at the top of the page insert the city you want to go to.
https://www.bonappetour.com/tony/a-t...e-vatican-city

For example, eating in homes in Venice.
https://www.bonappetour.com/s/Venice...31551509999997

Walks of Italy has them. I recommend you look at the video on this website.
https://www.walksofitaly.com/rome-to...-cooking-class

In Rome there is Eat With. They range from eating in the homes of Michelin chefs, to regular expert locals, highly screened cooks. They only accept 4% of applicants who want to host home cooking.

http://www.eatwith.com/list/italy/Rome/

http://www.eatingitalyfoodtours.com/...es-with-nonna/

https://www.cookening.com/en/explore/Italy

Since you mentioned you also want to go to small town, consider a cooking class at an Agriturismo, such as Fontanaro, in a small town in Umbria. It's a working Slow Food farm that grows its own olive oil, makes its own wine, grows its own food. You can take a day class, swim in their pool, eat spectacularly. Or you can stay overnight in this small town farm paradise and take more than one class.


http://www.slowcookingschool.com/the-school/