Bologna, Modena, Parma

You won't find one factory that produces all three.
You will need to visit 3 different places.
Vinegar factories (Acetaia, pl- acetaie) below:
https://www.visitmodena.it/en/discover-modena/land-of-flavors

Parma is charming and, I believe, the most affluent city in Italy. Not too big or too small. All three places very easily connected with trains. This part of Italy (Po Valley) is very flat...like parts of the Midwest.

All three cities could be a place to stay to visit the area. Parmigiano-Reggiano cheese is produced in the whole provinces of Parma and Modena (and Reggio Emilia, in the middle), and part of the province of Bologna. Traditional Balsamic Vinegar is produced only in the province of Modena (but there is a "brother" produced in the province of Reggio Emilia). Prosciutto di Parma is produced only in part of the province of Parma (but there is a "brother" produced in the province of Modena).
Reach the producers of Parmigiano and Prosciutto only by public transport could be very difficult and time consuming, because are mostly secluded in the countryside. For Balsamic Vinegar is easier because there are producers even in the center of Modena (including the Townhall).
If you don't want to rent a car and drive by yourself you can do a private tour with a guide and a driver, but isn't cheap. In this case I suggest Emilia Romagna Tours. They have even a group tour starting every day from Parma. Another solution is to directly book the producers and use a taxi, if are not too far from the train stations.

Thank you Roberto for the site. I had previously found a site but not to this extent.
Deebaxster and Ricky for your input. I just have to figure out which city to stay in. I know I will have to pay a premium by not having a car.
Previous to these cities we will have driven around Sicily for 16 days so Hubby will be done with driving.

There are day tours from Bologna to Parma and Modena.

Check out Bologna's official tourism site. It's been over three years since we visited, but we found their Bologna office to be very helpful--both with visiting Bologna and with the surrounding areas including Modena and Parma. We found it easy to take a day trip to Parma from Bologna; it was about an hour by train. We didn't visit any food-producing farms or factories, but I recall that tours could be reserved through the Bologna Welcome office.

I don't believe the new RS books include any information on Bologna; not sure about Parma or Modena (my RS Italy book is the 2019 edition). I found helpful information in the DK Eyewitness Italy guide (local library), as well as an Insight Guides book for Bologna (Barnes & Noble).

Our upcoming trip will have us in Bologna for 5 days, seeing that city and day tripping to the coast (Ravenna) and no further west than Modena. This is followed by 4 days based in Parma, seeing that city and day tripping east toward R-E, and points south and west.

Rick Steves' guidebooks tend to concentrate on cities included on his tours. I've used Lonely Planet for these locales. I have to say, they just aren't written as interestingly as the RS books.

There is an all-day food tour by Italiandays.it that covers all three of the food experiences you are seeking. If you stay in Bologna they pick you up at your hotel. I enjoyed the tour very much and came home with some fancy vinegar.
Cynthia

I experienced a tour with @www.italiandays.it....FANTASTIC! learned so much about Balsamic, Parmesan and Prosucitto...a very fun day out of Bologna. After, we were served an amazing dinner on site. A trip highlight with Allesandro as the best tour guide you could imagine.

We used www.italiandays.it this past June and had a lovely day. We learned a lot about Parmesan cheese making, actually going into the factory to see the process. We also had great insights into balsamic vinegar, and the different quality levels thereof. Our visit to the prosciutto factory was good, but not up to the level of the other two stops. At this venue, Alessandro handed the tour over to an assistant, whose English was not to the level where he could answer questions.

Aside from that, it was a wonderful experience, with two generous meals included. And our first time having wine with breakfast! Alessandro is great, very entertaining and very knowledgeable, an excellent combination.

We were based in Bologna, and the driver picked us up just a couple of blocks from our hotel, and returned us even closer. I definitely recommend this tour.