I have an ex who lived in New York a few years ago and so got to travel there quite a bit in the 3 years we were together.
Central park is good, it is zoned with different 'themes' for different areas. It does get busy though.
My Ex lived in Queens for part of the time and it was a pretty nice and safe area when I was there (Forest Hills on the MTA
map). She then moved back in to Manhatten around 96th st East and I also visited one of our friends in Harlem. Out of those places, Harlem was the most dodgy looking but I had no problems travelling alone in the early hours from Harlem on the Subway down to the bottom of Central park and walking along the bottom of the park (on the street, not in the park) and then on the MTA back up the East side of Manhattan (there was not an easy crossover between the blue and green lines north of the park).
Battery Park was a fairly nice stroll and you can walk up the East side along the river. There is a shopping centre (next to lightship Ambrose) after the ferry jetties. Just on the other side of it you can get a fantastic view of the Brooklyn Bridge.
Another sight which may be worth a visit is
USS Intrepid an old WWII aricraft carrier turned in to a museum. They also had one of the decomm'd Concordes there when I visited.
Greewich Village is great if you want to be around the student type 'candles and knitted hats' crowd

.
The Museum of Natural History was good but I think we stole most of the interesting things from around the world first so I still prefer the Natural History or the British Museum in London. The Guggenheim was ok but not my cup of tea really.
Don't forget there is more than New York.
Why not try
Bear Mountain. There are coaches which go from Port Authority IIRC.
Google Maps estimates 49 minutes by car from Manhattan. I stayed over night there at the
Bear Mountain Inn with my ex and we did a fair bit of walking. If you like hiking and nature then it is great. Be aware though that lots of people decend there around the lake for camping at the weekends and leave the place covered with litter.
Atlantic City is also just under 3 hours away by car or coach. Stay overnight and enjoy the Casinos and shows.
Coney Island is aroudn 30 minutes away and Long Beach is under 3 hours.
Lots to do and see.
Enjoy your trip
RB