why are people saying LIRR, he said hampton inn, which is a hotel in the city, not long island. Take the subway or a cab to get around, never rent a car in the city, parking is too expensive.
He said Rockville Centre area in the op. Being that you're located on LI, you should know that Rockville Centre is a nice little town on Long Island with a stop on the Babylon Branch of the LIRR.
To the OP:
Use the LIRR from Rockville Centre to Penn Station to get into the city.
Buy the ticket before getting on the train. From Penn Station, you can get to most places pretty easily by some combination of walking or subway.
I'd stay away from Times Square - totally overrated tourist trap. There are all sorts of awesome museums in the city (Intrepid Air and Space, American Museum of Natural History, the Met, etc...). Central Park is a great place to walk around and some stuff to see in there as well. If you want to see the Statue of Liberty, it's probably better to see it from afar, either from Battery Park or by one of the previously mentioned water options. I don't know if they still do tours of it, but I remember the UN Building (as lame as the UN is) was actually kind of neat on the inside - lots of artwork and architectural design to the main chambers. There is also the standard walk across the Brooklyn Bridge, go to the top of one of the buildings in Rockefeller Center (Top of the Rock; I hear the view is much better than the ESB).
Edit:
In regards to the LIRR, it looks like your hotel is within a 5-10 minute walk from the Rockville Centre LIRR station. If you're going to be going into the city for a few days, you can buy a 10-trip off-peak ticket (good for non-rush hour trains; you'll need to pay an extra buck or two if you get on a rush hour train with an off-peak ticket) to save some money and just have the conductor punch it twice each time (once for each of you).