NJ transit main line train service goes to secaucus junction to switch to a train to NY penn station midtown or continues to hoboken where you can get PATH trains to lower manhatten
For train schedules go to www.njtransit.com and put in Mahwah Station as your starting point and "New York Penn Station" as your destination (not Newark Penn Station). This gets yo to 34th street and 7th Avenue a very convenient location to see everything in NYC - here's a map of NYC showing where Penn Station is http://tinyurl.com/px9654p
Mahwah Station is here http://tinyurl.com/pq7ogsc and they have limited parking
I don't think they have bus service from Mahwah to NYC
But are you renting a car for use when in Mahwah? If so then you could also drive into NYC and we can give you advice on that. Let us know if that's an option (though the train would be easier assuming the schedules work for you)
Note that you can also call NJ Transit and speak to an operator during business hours (973) 275-5555Edited: 23 July 2014, 16:13
Newark is the only airport in that area of NJ. As for site seeing there is plenty of sites in NJ as well. There is the Ramapo Reservation right in Mahwah as well as the Les Paul Meusem. Here are a few links.