I’d second MoMA, something for everyone in there, it covers a huge range of modern art. The MET is awesome too. I got totally lost in there. The Frick is also worth a visit. Tiny compared to the MoMA and the MET but well worth a visit.
The Highline is really good for a walk. When you’re at the bottom end it’s worth heading down to Gansevoort Street. Lots of nice shops, including Rapha if you like that sort of thing.
I love the area around Houston and Bowery too.
If you want a good view of the Statue of Liberty but don’t want to pay fr a boat tour, the Staten Island ferry is free and leaves every 15mins/30mins depending on the time of day.
Oh, and walk over the Brooklyn bridge and checkout Dumbo, cool atmosphere and good views of Manhattan.
In the summer Smorgasburg in Williamsburg is a must do, it carries on indoors over the winter. Worth a trip for really good flea market style food.