Hi there. I've been in and out of La Guardia many times, so I'm quite familiar with transportation options:
$1.50 gets you a ride on the M60 bus, which makes a loop through La Guardia and then heads into Queens and Manhattan, making stops near many subway lines on its way. I think you have to pay another $1.50 to get on the subway unless you buy a Metrocard -- not sure if they sell those at the airport. The closest subway to NYU medical center is probably 33rd street station (at Lexington Avenue) on the 4/5/6 line (watch out -- only the 6 train makes that stop -- the others are express.) From there, NYU medical center is less then a ten minute walk. Here's the website of the MTA (New York's transit system), which has plenty of information and maps:
http://www.mta.nyc.ny.us/
The bus/subway combo is your cheapest option, but it involves climbing up and down several flights of stairs, which can be annoying if you have heavy or awkward baggage. Last time I travelled by subway from LGA, it took me about 45 minutes to get downtown.
$10 gets you a ride on New York Airport Service's minibus, which travels to Grand Central Station and then on to Port Authority. The ride from the airport is usually about half an hour. Get off at Grand Central (42nd and Lexington) -- from there, you should be able to walk to NYU medical center in twenty minutes.
$13-ish gets you on a shared-ride shuttle operated by one of two companies. This would take you to NYU med's door, but you might be making other stops first, depending on what route the driver follows and how many other passengers there are.
Taxis are around $30.
To get to any of these vehicles, follow signs for Ground Transportation when you get off the plane.
Safe trip, and have fun in New York!