One of my top picks for New York City is The Meetles.
The Meetles are a Beatles and 60's tribute band, playing in Central Park and other spots in the city. Their music is incredibly good, probably better than any paid concert you could go to.
You can read about the day I first stumbled on The Meetles in Central Park - it will give you a good idea why I plan to seek them out on my future NYC visits.
The Meetles play weekends on the subways, and sunny Sundays in the same spot at the 72nd Street entrance to Central Park (across the street from the Dakota, near Strawberry Fields). You can find their schedule on The Meetles Facebook page.