MOMA is excellent. Just beware all the selfie sticks hogging the front of Starry Night. I already mentioned what's on at the Met.
If you want to go a little ways out from the middle of Manhattan, the Met's Cloisters Museum at the very north end of the island is lovely. And that area should be beautiful this time of year - it's in a park right on the Hudson. But YMMV - depends if you like Medieval art.
One of my friends does the tours at the One World Trade Observatory, so I'll put in a plug for that, too.
The High Line and new Whitney Museum are beautiful, but both will be mobbed.
I will also add - it's a New Yorker thing to wait in line for food fads, such as cronuts. I don't get why, though I've lived here for years. If I were a tourist with limited time, I wouldn't do it. On a similar note, Shake Shack is perfectly fine, but also not really worth waiting in line for (the Times Sq line is crazy). I've had better burgers at more of a sit down chain called Five Napkin Burger. (However, if you go to Chinatown and see a line for a hole in the wall type restaurant, it's probably worthwhile.)
If you like mac and cheese, go to a place called S'mac, but not the one in the East Village. The one on 33rd st is less crowded, and not far from Empire State.
I could probably add a lot more, but that's all for now!
