Oooh yes @Sheena the HighLine is gorgeous.
1) Central Park. So many walks and 99% is free
2) Staten Island Ferry, free, goes past the Statue of Liberty (and it's a great walk to get there too)
3) Brookyln/Williamsburg, super cute, super hipster, some great cafes over there. Walk the Brooklyn Bridge to get there, the walk is 100% worth it! It's far though so plan some time in the day for it.
4) Times Square at night, even at 11pm there are still huge crowds there and there's always something going on