Everyone has given you such good suggestions. Here are my contributions.
Go to 59th Street and Lexington Ave. and you'll visit Bloomingdale's! Walk to 57th Street and keep walking. Some of the best (and most expensive) stores in the world are on 57th including Bergdorf Goodman and Tiffany & Co. You can pretend that you are Audrey Hepburn.
The Russian Tea Room is also on 57th Street. Try to take a carriage ride through Central Park which begins on 59th Street. Check out the zoo although it isn't as good as it used to be.
The Empire State Building has the most incredible view. On a clear day you can see for miles.
Go downtown and take the Circle Line to Ellis Island and the Statue of Liberty. Wear comfortable shoes because you will climb stairs for half an hour if you want to reach the top.
The Bronx Zoo!
The museums and theatre in New York are world class. The Producers, Les Miserables, and Cabaret are still hot tickets. The Guggenheim, Museum of Modern Art (MOMA), and the Metropolitan Museum of Art should not be missed. Also visit Lincoln Center.
I'll try to think of some more places.
Regarding wardrobe, June can be brutally hot so be prepared. If you will be in doors, most places are air conditioned and can feel like the Arctic. I guess it isn't that different from Southern California.
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