Several great museums in the City where you can walk around (w/o worrying about the weather), engage in light conversation in a relaxed and quiet mode and where a few moments of silence wouldn’t be awkward and where both of you could react to the subject matter and thereby learn a bit about the other. The Metropolitan Museum of Art, the new and expanded Museum of Modern Art, Hayden Planetarium (with the sky show), Museum of Natural History and the Jewish Museum are the more obvious venues. If its a nice day, the Bronx or Central Park Zoo or the Science Center in NJ are other options.