When I land in a new city, I spend the first day with no plan, just wandering about. It's surprising how much of a feel you can get for the texture of daily life in a city by doing this.
From my perspective, food and all the rituals and customs around it are some of the richest parts of culture, so I also like to go to grocery stores like this article covers. Another half-day is dedicated to a food tour, which is a fantastic way to learn about a new culture. In my experience, often the tour guides have lived in that culture for many years but they're not originally from there, so they can do cross-cultural translation for you, pointing out things you might not expect. And the best tours also include walking around the city and seeing cultural highlights.