tkim,
public transportation in NYC is much easier than expected, even for a person who is not from NYC; with a map in front of you, it is somewhat self-explanatory. Nobody stops you, of course, from taking a cab if you're willing to spend just as much time (that's if you're lucky and you don't get stuck in traffic) and much more money than by using public transportation. (It is so "unreliable" that many times the Mayor himself uses the subway on his way to work). Getting back to your question...
Here is a map of the NYC subway:
http://www.nycsubway.org/maps/route/
The Brooklyn Mariott is near Jay Street-Borough Hall subway station (which, if you follow the blue line, is the 2nd subway station in Brooklyn after you left Manhattan). The Jay Street-Borough Hall subway station is a hub for most trains to most directions in all boroughs of NYC.
To get to Maimonides Hospital, you'll need to take the Brooklyn-bound D or M train (make sure you don't take the train in the opposite direction) from Fulton Street (this is within walking distance from Jay Street-Borough Hall). You get off the train at 50th Street (in Brooklyn). The hospital is located on 48th Street.
I don't know which AE exactly you're talking about? Are you talking about Montefiore? Because that is a long ride both by train or by cab...that hospital is in the Bronx!