All of the above, but spend time down town, TriBeCa, west village etc.
This is the low rise, red brick area, has an awesome feel.
Walk the highline (free) from north to south, this is free and just amazing.
At the end is an undercover area with cafés and bars.
The Boris bikes are good as you can pretty much do the circumference without crossing roads so it’s a nice ride. You can ride or walk the Brooklyn bridge (for free)which is on a deck way above the traffic.
Staten Island ferry is worth it as it is free.
Pizzas, from my sister who lived in the village:
Joe’s for a slice, John’s to sit down for a pie- both in Greenwich village. I prefer Joe’s. For a fun restaurant w good Italian pizza: Ovest in Chelsea.
I preferred Johns btw.
We went to a club at the top of the Standard Hotel, which was amazing, views of just about everything from the roof top bar. Obviously no chance of getting the kids in at night and you will need passports, not sure what goes on during the day:
http://standardhotels.com/high-line/food-drink/le-bain