The Brooklyn Botanical Gardens is beautiful when the cherry blossom is out. It was free on Tuesday's when I was there in April 2016. Walking from DUMBO along the waterfront to Brooklyn Bridge is a really nice walk though hardly a hidden gem. Lots of great bars in Williamsburg and Greenpoint. Get out and explore New York so many people just stay in Manhattan. The Bronx and Harlem are really interesting.
I would think that if you miss the main sights in Manhattan you will kick yourself.
It all depends what you are interested in but we loved the Highline. Its easy to walk all of it. I may be a bit cold in spring but it is a fabulous idea for an old railway line andyou can stop off at Chelsea Market. You must have a look at sme of the iconic buildings. Obviously the Empire State Building but Top of the Rock has great views too. Central Park is worth a visit and the Met Museum is full of great exhibits. The Radio City Music Hall surpassed our expectations. It really is a superb Art Deco building.
We also like MoMa and the Guggenheim but not eveyone will. The SoHo area is interesting and you can see the older buildings with external fire escapes in this area and also in Greenwich Village. These areas have narrower streets and a more neighbourhood feel. Great cafes. Go to Brooklyn and then walk back over the Brooklyn Bridge and then it is then easy to walk to the Twin Towers museum if you wish. The Staten Island Ferry goes past the Statue of Liberty and Ellis Island is really thought-provoking.
Then there is just the sheer size of the buildings. Grand Central Station interior is fantastic and you must try and gaze on the top of the Chrysler Building. We just do not have anything like that here. Times Square is a bit - average. Neon lights but not much else.
My advice would be to have a look at the Lonely Planet guide and you will get a real flavour of New York but tailor it to what you are interested in. They also have good recommendations for restaurants. We found the best places get very booked up though, so plan. Eating out is way more popular than cooking! I can recommend the Mercer Kitchen.
As a previous poster said, the high line is a nice walk, stop off at Chelsea market before/after for a browse and food.
We went up into the crown of the Statue of Liberty on a previous trip which was fun. They only sell a limited number of tickets for each day so you need to book in advance.
Keens was recommended to us for steak but we couldn't get booked ther. We had a lovely lunch at the boathouse at Central Park and rented a boat on the lake. Walk over the Brooklyn bridge and go for pizza when you get to the other side (if you can't tell we ate a lot!).
If you want to do the 9/11 museum its (understandably) quite emotional. We pre-booked for an early timeslot so it was empty when we got there and could take our time. By the time we left it was getting quite busy so I'm glad we went early.
You'll have an amazing time! I can't wait to go back with DD.
We are here now! Agree with the above posters, plus we really enjoyed hiring bikes in Central Park yesterday and allow a lot of time for Ellis island and statue of Liberty. One world tower at sundown is good too.
The two hour boat trip (circle line tours) was really good, as was cycling round Central Park, walking the highline, eating we loved The Spotted Pig in the West Village... You will have an amazing time!
We did a food walking tour which was incredible and a great introduction to lots of districts.
I loved Govenor's Island on a nice day. Cycling round Central Park. The cocktail bar on top of the Met. The Morgan www.themorgan.org/ library (great cocktails). Also Grand Central with the whispering corner and the Oyster Bar.