We spend a week in NYC every year. I have never had an issue carrying my gear around, especially if you're going to be in the usual touristy areas. At any given point you can look around and see a half dozen DSLRs hanging around people's necks. I bring my large backpack to get my gear to NYC, then I use a Crumpler 6 Million Dollar Home bag, with the wake-the-dead-loud velcro closure, to carry gear once I'm there. No one is going to get to my gear without my knowing it. I carry it cross body, so someone would have to cut it off me to get it. I don't use neckstraps for my cameras. I prefer to keep my stuff in a bag and use wrist straps when I have my camera out.

Central Park on a nice day. You can easily spend the whole day shooting and not get half of it. There are always musicians and artists and weddings and all manner of interesting people to capture. For people watching I also love Union Square. Grand Central Terminal is worth a look as is a trip to the Top of the Rock (Rockefeller Center) Great views of the city and Central Park from there. If you go to the World Trade Center area, you can buy a $5 r/t ticket to take the subway across the river just one stop to the New Jersey side and get spectacular views of the NYC skyline, then jump back on the subway and come back over.

And as someone else mentioned, B&H is not to be missed. As a Brit, you may want to put off any major purchases you might be considering until you get to NYC. I think you'll find the savings to be substantial over what you'd pay in the UK.