What to see: Everything you can--it is all amazing! My children always want to see the dinosaurs and the whale.
Extras: Try the food in the cafeteria, it's reasonably priced and pretty tasty.
Our favorite spot: The Hall of Minerals is sometimes quiet (if you need a moment to breathe). If you can, see the Easter Island Head, it is a great photo op. :)
To do before you go: rewatch Night at the Museum. My kidlets loved telling me things they saw from the movie in the actual museum (though most of it has been redesigned for the storytelling part of the movie).
Little known fact: they have a FREE toddler story time in the Discovery Room--with a theme from the museum. Afterward they take the children to see certain parts of the museum relating to the story they read. My children adored this and they have activities in the room for children to take part in--digging for dinosaur bones; hats from different regions; etc. Also, it is pay what you wish all the time. Thank you AMNH people!