Right outside of Columbia's gates is a 1 train stop and several bus stops. This means access to just about anywhere in the five boroughs. Public transportation is a wonderful resource, and the only sort of transportation you will ever really need. Though it can be faulty on weekends and late nights, the relatively cheap cost and predictability means that it is an ideal way to get around the city. For those who are in a hurry or don't feel safe riding the subway (a fairly groundless fear—subway stations have attendants and are filled with cameras), cabs are a much more expensive option. Not in a hurry? Try walking or taking the bus! It's slower, but you get to see the city and, as a result, get to know it better.
A resource which few students take advantage of is the evening shuttle. This shuttle has several routes and will take you all over the Morningside Heights area. It is especially helpful for evening runs to Fairway Market located in Harlem. This massive grocery store has great prices, and the shuttle makes the trek back to campus much easier.