You can stay with your dog at many pet-friendly hotels, many are glamorous, like the Waldorf-Astoria. There are also a large number of pet-friendly vacation rentals to choose. Many of these hotels have arrangements with pet sitters to come to your room to watch your pet, if you are going to out to a Broadway show.
Many of the famous stores in the city are also pet friendly, Bloomingdales, Saks Fifth Avenue, and Tiffany's to name a few.
The Brooklyn Bridge is a definite destination spot for visitors, as well as, residents of the city to check out with their dog. There are guided walking tours of lower Manhattan that welcome dogs. The Central Park carriage rides welcome dogs with their people. Since NYC is primarily a walking town, you and your dog will probably like that option very much and will enable you to see the famous landmarks, the Empire State building, Grand Central, Madison Square Garden, The new World Trade Center and the awesome New York skyline from the street.
NYC has more off-leash parks than any other city in the United States. The dog parks are free and no permits are required.
Enjoy NYC with your dog :-)