liberty science center in jersey city is cool, but not worth taking away from your NY trip to see. i love jersey, but stick to seeing ny.
My absolute favorite thing in NY is the Cloisters in Fort Tryon Park. It's a medieval art museum and the park has amazing views of the Hudson River and the NJ Palisades. The museum houses the very famous Unicorn Tapestries as well as a lot of other cool stuff. Take the A train uptown to the very last stop. It's part of the Metropolitan Museum of Art (another must see) and what I'm about to tell you will change your whole trip....
you and your wife can pay a dollar each to get in. or less.
why? because admission fees at the Met are considered donations, and they only "suggest" $15 or whatever the sign reads. Me and a friend went to the Cloisters last week and paid $5 for the two of us. The person selling tickets will not judge you or even flinch.
After you go to the Cloisters, which is quite small and does not need a whole day, go and check out Harlem. You're already there. Go to Sylvia's Restaurant Sylvia's Restaurant at 328 Lenox Ave, NY, NY. Phone 212.996.0660. If not, head to museum mile and go to the Met, you've already paid your dollar! Just hold on to the little "M" pin they gave you at the Cloisters (a fun little souvenir too).
If you feel like staying in Fort Tryon Park, eat at the New Leaf Cafe. The singer/actor Bette Midler owns it and your meal helps fund green spaces and playgrounds in NYC. The food is great and they make a drink with bacon infused bourbon. It might just be on the brunch menu, but ask for it anyway.
My favorite Greenwich Village restaurant is Dojo at 14 West 4th Street, NY, NY.
http://www.njtransit.com is also a super helpful website for bus and train schedules.
bon voyage!