Well worth visiting the 9/11 museum. It has such an eirey atmosphere. Really interesting too.
If you want to see a broadway show, there is a ticket booth in Times Square that sells tickets for shows that haven't sold out for up to half price. We saw school of rock, would recommend it!
The skyline picture was taken from just over Brooklyn bridge. There was a fantastic pizzeria called Julianas.
If you want to see the Statue of Liberty without actually going to the island, there is a free ferry to Staten Island which passes the SOL. It's easy to find.
Soho and Greenwich are nice places to walk around. There are some actual pubs in these areas too. Wouldn't recommend Chinatown really.