Seeing how locals live and eat is probably one of the main reasons you travel. One way to explore the regional cuisine and find out what locals really eat, according to Anthony Bourdain, is to visit a local open air or grocery market. Within your first day or two at a new destination, hit up a market to get a sense of what locals eat and what's in season. When I did this at Kyoto's Nishiki Market, I was able to see everything from plump, colourful mochi to grilled baby octopus on a skewer to huge barrels of pickled daikon. My time exploring there inspired me to try those same things on local menus, like wagashi (sweets specifically made to be enjoyed with tea).
You can also use the market as an opportunity to stock up on snacks or ingredients if you plan to cook a few meals during your stay. Even if you're just there to look, take note of anything that you want to try. Write down the name or take a photo so that you can recognise it on menus later. Visiting a market is also a good way to ease into the culture at your own pace since you can go as slowly or quickly as you like and you don't necessarily have to talk to anyone to get a full experience.
How Anthony Bourdain Dives into a City [Men's Journal]