Getting to NYC:
You can fly into JFK, Laguardia, or Newark.
There’s an MTA bus (the M60) that drives from Laguardia and down 125th Street, which will get you to Columbia and the Aloft Hotel in around 40 minutes.
Here are directions for travel to Columbia University’s campus from each airport. Keep in mind that the directions will be slightly different if you’re heading to the hotel.
Buses are too numerous to list. Many arrive to Port Authority, and many drop off all around the city.
Getting around NYC:
Here’s a link to the MTA subway map.
The closest subway stop for Columbia University is the 116th Street Station on the 1 train.
The closest subway stop for Aloft is the 125th St station on the A,B,C, or D trains. It is also very close to 125th St on the 2,3 trains.
The 125th Street station on the A,B,C,D lines is an express stop. Taking it will get you wherever you want to go (downtown, to Brooklyn, etc.) easier and faster.
Many people use Google Maps for subway directions, and its timing is pretty accurate.