Try The Frying Pan on Pier 66. The food is amazing. Since you're going in the fall, call ahead for hours (they're open daily, weather permitting) and make sure you bring a sweater since it's on the water! Here's their site: http://fryingpan.com/site/
That's easy; any local would point you towards Mert's Heart & Soul. It's decently priced, exquisitely prepared and completely authentic soul food. If for some reason that doesn't float your boat, there's always The King's Kitchen on Trade Street, which also boasts a stellar reputation and is beloved by comfort food connoisseurs from all around.
Two theatres immediately spring to mind: the Actor's Theatre of Charlotte in Uptown and the Belk Theater on First Ward. Both provide some stellar performances, with the Actor's Theatre slightly eeking out a superior reputation. Either location will do, and both can promise a well-staged performance that's well worth your money.
It's too bad, when I was younger I was actually able to climb to the crown of the Statue of Liberty, but they don't let you do that anymore. Ellis Island is still open for visiting though, if you want to get your share of history. Other options are Times Square for shopping and broadway shows, Coney Island during the warmer months, and definitely a walk along the Hudson, no matter the season.
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