I would definitely second the idea of buying one of the passes, as they massively reduce the cost of doing multiple things. Shop round, though, as some offer discounts if you book bus tours at the same time etc., and the pass prices can change.
Have you booked flights to JFK or Newark? If you are flying to Newark, staying in Jersey City would save you a lot of money. As someone has already mentioned, make sure you are near a PATH station. This will make it easy to get from Newark airport. It is also only a quick hop into the financial district or about 30 mins to 33rd Street (with pptions to get off at several points in Lower Manhatten). I can thoroughly recommend Grove Point. I think Furnished Quarters own the rental apartments there. They are large, very well kitted out self-catering apartments directly above the Gove Street PATH station.
In NYC, I'd definitely consider going up either the ESB or the Rockefeller. If you do the latter, you get amazing views of both Central Park & the ESB. State Island ferry is free. There is also a really cool cable car type thing that goes from manhatten to Roosevelt Island and only costs the same as a Metro trip fare. On weekend during the summer. (Not sure about during the week), there is also a free ferry to Governors Island, which leaves from Battery Park, and they often have event on the island. The World Trade Centre memorial centre is free, but book in advance. Also it's worth spending lots of time wandering round places like Chelsea, Grand Central, Fifth Av., We also found that some of the kayak places in Brooklyn offered free 'have-a-go' sessions early in the day and late in the evening, so have a dig around online.
As for climbing the Statur of Liberty, it isn't too busy as tickets are limited and timed. There are small platforms at various level, which enable you to stand aside and let others past, but it is still a bit of a conveyor belt and you don't really get much time at the top. If you do book tickets, make sure you give yourselves enough time to stop at Ellis Island and look round the museum. I believe boats only stop on the way out. Also, if you do stay in Jersey City, there are boats from Liberty Park, which can be reached by either the light railway or by a boat that crosses the canal.
Have a good trip.