I've just got back, and done the usual touristy things :-
Empire State Building (at night - superb views). Didn't have a ticket, and took around half and hour to queue and go though security.
Top of the Rock (day time - you can see Central Park and Empire State Building). Much more relaxed than the Empire State Building and you can sit around up there for as long as you wish. Didn't have a ticket, and only took 10 minutes in queue.
3 hour river cruise with Circle Line Cruises. A lovely close up of the Statue of Liberty and the manhattan skyline.
A rickshaw in Central Park. $90 for an hour, but my mum has walking difficulties so it was our only option. If you are able, hire a bicycle and go on the weekend where no traffic is allowed through the park.
Staten Island ferry. Free of charge and and you get a pretty good view of the Statue of Liberty.
9/11 memorial. Very moving and should be seen. You can't imagine the sheer panic and devastation that day caused. I booked tickets (although free) online before I went, but you can queue on the day.
Times Square - just walk around at your leisure
Went into Macy's, although didnt make a purchase!
Lots of places to eat, especially if you like burgers, steaks, pizzas, etc
Metro pretty easy to use as is the bus. You can't pay cash on bus (or train) so get a ticket before you travel. $30 gets you unlimited travel on subway and bus for 7 days. I am sure this doesn't include the Long Island Rail Road (LIRR) which we got to and from air train to airport. Was easier for us with mum and her walking trolley.
I saw mamma Mia at Winter Gardens Theatre. I bought my ticket online for $90 so I didn't miss out. The person next to me paid $75 on the day. Just giving you an example of a bit of money that could be saved on tickets, but I wanted to make sure I saw the show and really wanted to go on the Sunday night.
We took a train to Washington, D.C. for the day and that was great.
I felt very safe in NYC and didnt once feel threatened or intimidated. We had a wonderful time.
Sorry if I have waffled! Feel free to ask anything and I will be only too pleased to help. Have fun!