I went to NY alone a couple years back and had a fab time, is this just you or is it a family trip?
If it's just you, don't be overly concerned about safety (I was given some right horror stories when I started to tell people I was going alone.) Stick to the main touristy areas and be sensible and you'll be absolutely fine. I got around using the Big Bus company as they drop you off right outside all the major interest places and was worth the money IMO to be able to avoid the subway. But all depends how confident you are and who you're going with.
Definitely recommend taking a boat trip out on the river to see the Statue of Liberty. Also, you can go inside the One World Trade Centre and go right up the top, views of the city are just incredible, highly recommend you do that too.
I'm not a big shopper to be honest but there are tons of gift shops and some fun places to go in especially if you're with children, there's the Disney Store and the M&M store in Times Square, both worth a look.
Do avoid the costumed characters outside the Disney store though, they are known for trying to get people to take a photo with them and will then demand money, best to just ignore them. I don't believe they're actually officially with the Disney store either, just random people in suits.
Oh, Central Park is wonderful, it's a really beautiful spot right in the middle of the hustle and bustle. Also recommend the Natural History Museum which is right alongside, if you're in to that kind of thing!
Have a nice trip!