Starbucks is by far the dominant coffee shop in NYC – there are well over 200 stores in Manhattan alone and many times you will see two Starbucks’ across the street from one another. The next runner up is Dunkin Donuts.
That said, we’d really encourage you to visit independently owned coffee shops instead, at least while you’re visiting NYC. You can have a Starbucks or DD anywhere, but NYC has some amazing coffee shops – including some smaller chain coffee shops. I’m personally a fan of places that have only one or two locations, but some of the smaller, great quality chains (2-10 locations) include: Stumptown, Joe, Blue Bottle, and La Colombe. The downside, of course, is that some baristas at these types of places can take themselves a bit too seriously, though even then, that usually results in a meticulously prepared beverage. But generally all of these have a cool, relaxed, bustling vibe, with more personality than your large chains, and better quality, fair practice coffee.