Good advice above. NYC is definitely worth a few days' stay, as there's so much to see and do. The area immediately around the pier is nearly desolate, so I also suggest staying in the Times Square area. So is the area just to its north, bordered by Avenue of the Americas (Sixth Avenue) and West 57th Street. I have heard of a couple of hotels near the pier, yet don't have any first-hand experience.
One thing about New York--once you park your car, you won't need it again during your stay. (Bring comfortable shoes for walking.) Some hotels may offer a cruise-and-park package, yet it will cost $$. There's also parking at the cruise terminal for $30 per night, which is a bargain for NYC.
As for attractions, Times Square is cool, as is the lower part of Central Park. I was in NYC just a few weeks ago, visiting friends who were about to embark on a Bermuda cruise, and we took the three-hour Circle Line boat tour around Manhattan. It was fabulous--and this comes from a native New Yorker. I can't recommend it highly enough.
I've cruised Carnival out of New York a couple of times and found it very organized.