Just spending two weeks at NYC, a few quick tips below:

1. Take NYC Bus Tour of Brooklyn, Bronx, Queens.

2. Go to nps.gov to see a lit of National monument inside NYC:

• Federal Hall National Memorial.
• General Grant National Memorial
• Governors Island National Monument
• Hamilton Grange National Memorial
• Castle Clinton National Monument....

?3. Visit Governor island and roosevelt island. Take roosevelt island tram.

4. Attend tours: I recommend freetoursbyfoot.com. 

Just for your info. It is totally my suggestions.  I understand everyone has different way to travel. 

I read travel post at this forumn very often, just want to share back.