There's this new book out about NYC that I think is essential for anyone who wants to see the city on a budget. It's Frommer's NYC for Free and Dirt Cheap. It contains TONS of good information about where to go, get and see free things.
I've seen a bunch of free concerts, taken discount ice-skating in summer, gotten in to broadway shows for half the price etc- but I had to get the info from almost random sources. This book puts it all together for you, so all you have to do is flip to the page and plan your trip.
It's the second edition and costs about $14. I suggest you buy this edition, since some of the places and things in the first edition are no longer available.