While not from NYC I can tell you safety should not be a concern. I took my teenage daughters last summer and had a blast doing that tourist thing.

I agree with a previous poster that you should buy a Metrocard. Info here. Also study the subway system map prior to your trip. I printed it and carried it on my trip. Link here. The Metrocard is also good for MTA buses. Google Maps on my smartphone was a great tool to help get around the city using public transit.

Have fun and let us know about your trip!