NEW YORK CITY (MANHATTAN) A SAFE PLACE FOR TWO TEENS?

Hey, So This summer my friend and I (she's a girl) are planning to go to New York City in August as her mom is doing business over there and she wants a friend to accompany her so she isn't alone the whole time. So I was just wondering if Areas like Downtown Manhattan, Greenwhich Village, Soho, and Noho are safe places for us to be roaming around, exploring & sight seeing and doing fun stuff alone? We are both fifteen years old and are going into grade 11 as of Sept. 2011 and I think that everything will be fine as long as we play it smart. Example: Don't go to any sketchy areas such as Harlem, Brooklyn, or Queens, (we will be on the island) or the Bronx. / Definitely not walking through central park at night, and Just being back at the hotel by a good time like 8-9 or something. I have always wanted to visit NYC and I really hope this works out as does she so if anyone can help me out and Answer this that would be awesome! Also If there are any MUST SEE things in Manhattan for us to see let me know. ( Chrysler Bldg, Empire state bldg, Statue of liberty, Rockefeller Center.. etc.) THANKS :)