If you want to dive into local cuisine and culture, one of the best things you can do is take a cooking class when you travel. This one was organized by our tour guide in Genova, Italy, at Creattivando (if you go, tell Mario and Cristina hi!). We made pesto in a mortar and pestle, handmade pasta, and a few other dishes.
As we were led through the preparation process, we learned about Italian meal traditions (many nonnas still cook a big Sunday meal, but they take more shortcuts now) and cuisines.
I left, sated, both from the food I helped make and then eat, and from the cultural immersion.