Modena

In the center of Modena you can visit the "acetaia comunale", the one inside the tower of the Townhall. This: https://www.visitmodena.it/en/discover-modena/land-of-flavors/the-vinegar-factories/municipal-balsamic-vinegar-factory-of-modena
Just outside the historical center (so a walkable distance) there is Acetaia di Giorgio.
For tastings I suggest the "Consorteria" shop in piazza Mazzini (almost in front of the synagogue), dedicated only to Traditional Balsamic. They represent a group of more than 20 producers, so you can do a comparative tasting.
All other producers are in the countryside, but someones not far from Modena so you can go even by taxi.

Parmigiano-Reggiano cheese is produced in the provinces of Parma, Reggio Emilia, Modena and part of Bologna, so dairies are located in these provinces. Is difficult have cowsheds near the center, so the producers are quite difficult to be reached without a car. You can check for one near the center (of Modena or Parma, as you prefer) and go there by taxi.
In the Consortium webpage there is a list of dairies available even for visits: https://www.parmigianoreggiano.com/dairies-visit-tasting/

Our guide from Tours by Locals drove us to places near Modena that produced cheese, balsamic, and wine, then to lunch --- he probably could have arranged tastings if we'd wanted them. Here's the website: https://www.toursbylocals.com/Italy-Tours

We have researched several tours available for the countryside around Parma, Modena and Reggio Emilia. There are a variety of options depending on your wants. I think we even found a tour that offered balsamic tasting combined with a Ferrari tour!

Part of our "Italian Days" tour was breakfast and tasting of Several Balsamics of varying age up to 35 years. The other stops were at a Parmaigiano plant to see the process and also Proscuito. 10 hours of eating (and drinking) - wow.