We’ve had lots of threads here about places to go and things to do in NY. You might want to check them out.
If you want to see some of the less touristy, but still really fun and interesting parts of the city, come and spend some time in the Bronx. We’ve got a lot of great stuff to see and do. Check out the New York Botanical Garden and the Bronx Zoo. (Granted, those get lots of tourists, but their mainstays aren’t the tourist trade, unlike Radio City Music Hall or the Statue of Liberty.) You could take a very free-form, self-guided tour of the borough by hopping on the Bronx Trolley and checking out the different places the trolley stops at.
If you love to eat, make sure to visit Little Italy in the Bronx. It’s a 5-minute walk from my place to Arthur Avenue. Let me know if you’re thinking of coming to Little Italy, and I’d be happy to take you out for tiramisu at Palombo. (The place isn’t on the list in the Wikipedia article. It opened in, IIRC, January or February of this year. It seems to me that the Wikipedia page hasn’t been updated yet. But Palombo, IMHO, has the best tiramisu in the neighborhood, which is really saying something!) If you like fine wines, make sure to stop at Mount Carmel Wines and Spirits. I’ve become spoiled rotten by having the place as my neighborhood wine shop.
Also, if you like the outdoors, spend some time wandering throughPelham Bay Park. The place even has its own beach! I haven’t been to the beach yet, but I’ll bet it’s nice. (I haven’t been living in the Bronx for very long.)
I also suggest you take a trip to amazingly cool, multiethnic Queens. I’ve heard Astoria’s certainly worth a visit (I haven’t been there myself, yet). Definitely check out Jackson Heights, which has New York’s Little India. You could spend some time just gawking at the gorgeous, ornate, hand-worked gold necklaces in the Indian import jewelry stores. Oh, and you can see silk saris, some of which are so finely made that you can pull them through a wedding ring. No, I’m not making that up.
There are an awful lot of interesting museums in Queens. If you go to Jackson Heights, I strongly suggest you have something to eat at Kababish, on Broadway. There’s also a great Indian-Chinese place I went to in Jackson Heights recently, but I can’t remember its name right now. If you’re interested in hearing what that name was, let me know and I’ll send it to you or post it here.