New York Subway
Subway is the suggested method of transportation for any New York City traveler no matter the destination. The #1 and the #9 trains pick up and drop off at Columbia University Station. Once on those trains, you can access just about any point in the city. Subway maps are color coded and easy to read and are located at every subway stop and on the trains themselves. Contrary to what you may have seen in movies, it's okay to ask for help or directions from other riders on the subway, although many of them may be tourists themselves.
The M5 bus picks up and drops off on Riverside Drive at the West 114th Street Station. Once on the bus, you can traverse the entire city as long as you know when and where to transfer. During the day the bus stops several times an hour so if you miss one, you're only a few minutes away from catching the next one.
For many native New Yorkers, taxicabs are a standard operating procedure. If you want to guarantee service immediately, it may be helpful to call ahead and request a cab, although in the area surrounding the university, cabs drive by on a very regular basis and it's not all that difficult to hail one. This method will be considerably more expensive than the subway or bus, but if money isn't an issue and nerves from riding public transit are, this may prove best.
Seriously, why not ride a horse-drawn carriage? The university isn't far from the northwest edge of Central Park and empty carriages come by regularly. Depending upon how far you're going, this could be a viable option. And how often do you get to ride in a horse-drawn carriage anyway? Ask the driver to take you through the park and enjoy the ride. Caution: This method will be incredibly slow and only works if your destination is somewhat close to Central Park. If you are visiting with your loved one then this will be a fun way to get around.
Manhattan may seem huge in the movies, but it's not that difficult to walk around it. Again, this option only works if you're not in a hurry and if are staying within Manhattan. But if you want to really take time to see the sights, a walk would do nicely. As a point of reference, it's about an hour and a half walk from Columbia to the Empire State Building.