I take my family to New York two or three weekends every year. We usually stay in the Times Square area for one of the weekends. I highly recommend you do this but it can be quite pricey over the holidays. On the other weekends we usually stay at the Embassy Suites in Secaucus. Free parking, free breakfast, nice pool, and an 11 minute bus ride to the Port Authority at 41st and 8th. The bus is about $2.50 each way and the bus pickup area is about 50 yards from the front door of the Embassy.
If this doesn't work, I sometimes drive to the Hamilton New Jersey train station. It's just north of Trenton, but unlike Trenton is safe, easy to find, and has ample free parking. The slower train, New Jersey Transit, takes a little longer than Amtrak but costs about $12.00 per person and seniors/children are under $6.00 each way.
In all of the above cases you'll be in Mid-Town. Find a subway and get yourself to the Bronx. It's pretty far away and you'll want to look for the express trains in order to bypass some of the Harlem stops.
I've been going into the city since I was a youngster in the 70's and I know my way around. Ask any questions and I'll try to answer.
Since you'll be in the Bronx, I'll let you in on a secret. The Bronx has a Little Italy that hasn't been taken over by tourists like the one in Lower Manhattan. It's across the street from Fordham. A long walk to/from Yankee Stadium but a short cab ride. It's the best Italian food in New York and it isn't close. Here's a link:
http://www.arthuravenuebronx.com/
