What should I avoid
I would think your brother would be better able to answer that question.
, and more importantly what would be cool places worth checking out?
That depends upon what your preferences are (street, architecture, people, landscape, etc.). NYC has it all. A boat tour of Manhattan will give you great shots of the bridges from the water--a different perspective. A walk of the Brooklyn Bridge can give you many great shots of the bridge. Central Park can occupy you for days. I was there during Memorial Day one year, and I couldn't shoot fast enough! Consider taking shots (photos that is) from high up in nearby buildings to get a different view than a street view. If you have a telephoto lens (200 mm should do), take the Staten Island Ferry (it's free) and get shots of the Statue of Liberty as you go by.
I will have access to tripod as my brother is also into photography, I do want to get a night shot of Times Square while I'm down here.
I found it hard to get night shots because by the time I got set up, some rent-a-cop came by to tell me no-can-do. I knew I had a right to take the photos, but being a guest, I moved on. I had the same experience photographing buildings in mid-town and downtown during the day. Being prepared, courteous, and friendly might get you the shot anyway. A night shot of the fountain at Lincoln Centre would be great.