MUSEUMS THAT MUSEUM GO-ERS SHOULD GO TO, THAT AREN'T MOMA:

METROPOLITAN MUSEUM OF ART. There are no exceptions. You can't go to museums here and not see the met. It is amazing. And ranges from free to $5.

The Frick Collection.

The Morgan Library

The Guggenheim. A must-go for modern art with actual substance and quality, you will evade the inevitable heaps of shit that rot in the MoMA.

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AREAS YOU MUST VISIT IF YOU HAVE EYES AND FEET ESPECIALLY EYES

SoHo, AND Chelsea, which are nearby to each other, and are a must must must must

Columbus Circle/Lincoln Center

NOT TIMES SQUARE

That is all. Enjoy.

Edit: NikolaiFyodorov, this is for you. Also note that my taste is a little more hipster and places like SoHo and Chelsea are cultural dens of beauty and modernity and art and wealth, as opposed to locations like the Empire State Building, or Times Square which are just 'wow look at how big these shiny things are' kind of wealth. That said, I'll second veen's suggestion of going to the Top of the Rock. Seriously, forget the State Building, and either way you're probably gonna pass through Times Square during your journey, and a pass-through is all you need.

Lincoln Center/Columbus circle are in my opinion far more extravagant than the Rockefeller center, though that is also awesome. For context, all of these places are actually only a few blocks from each other, in what is called Midtown. Columbus Circle is also right by the South end of Central Park. Going a little more South, lands you in the Rockefeller Center. Avoid a cab unless you can really afford it, and don't be afraid of the Subway. Keep us posted on your plans!