I'm a writer and programmer that has lived in New York for over ten years now. Some of the NYC things I like to do are seeing museum and galleries so I may be able to point out some good shows. I also like to go to see improv comedy- not a usual choice for people with a limited time to see a city- but if you happen to be interested there are some very good and inexpensive shows here.
The place that I enjoyed visiting the most was Tokyo. It's not fresh in my mind right now but I thought the modern architecture was amazing and I liked being immersed in a non-Western culture.