To be honest, almost all the streets and lanes of this food crazy city has something or the other that will feel like the "best in the city". It's really impossible to name the best place unless you take a scrutinizing survey (like WHO has done and made the city proud!). But to throw a few names, there is Vivekananda Park, Gariahat, Park Street, sudder Street, New Market area. If you're trying out street food, it would be a sin to leave out puchka. Us Kolkatans have feasted on this heavenly snack all our lives and the worst we've had is stomach upset. Have a digestive pill after, if need be, but do try it at least once!
Also, since we're talking food, if you like Chinese, one of the best places in India for the cuisine is a restaurant named Blue Poppies (Sikkim House). They have the best momos, dimsums, thukpas and noodles that you will ever taste. Another area for Chinese food is Tangra, the Chinese settlement in Kolkata which offers authentic (though inevitably Indianized) and cheap food cooked by Chinese people.