We've had a couple of threads about this before:
http://www.lcficmbs.com/ubb/ultimatebb.php?ubb=get_topic;f=10;t=000304
http://www.lcficmbs.com/ubb/ultimatebb.php?ubb=get_topic;f=10;t=001212
http://www.lcficmbs.com/ubb/ultimatebb.php?ubb=get_topic;f=10;t=001860
But, when it comes to New York City... there's everything. And I do mean everything.
What are you interested in? Museums? What kind? Restaurants? Pick a nationality (even Ethiopian!). Landmarks? Take your pick. You get the idea.
Generally, most of the touristy stuff is (not counting museums and the rather vast Central Park) in Midtown or below. From 60th Street or so south.
Up that way, you've got upscale shopping on 5th Avenue and Madison Avenue.
At 50th and 5th, you've got Rockefeller Center, home of NBC Studios. There's more shopping, of course. And St Patrick's Cathedral is across 5th Avenue. But the view from the Top is hard to beat. You can see it by going to Top of the Rock, their observation deck. Cheaper and much easier than going to the top of the Empire State Building, and you even get a lovely view of the ESB (and a peek at the Chrysler Building). Alternatively, one floor down is the Rainbow Room Grill. You can go up for a meal, or even just a drink at the bar - as long as you meet the dress code (jackets for men) and can deal with the prices. Still, a couple of drinks should be cheaper than the price of an observation deck ticket. Good news and bad news (depending on the weather and such) is that you're indoors, behind plate glass windows. The view is a little more restricted, but the place is air conditioned...
Going across to the West Side, you've got the theater district, which runs from there down to Times Square. Broadway & Off Broadway shows, and a variety of restaurants.
Then there's Times Square itself, which you just kind of have to see. It's touristy and glitzy and crowded. Giant neon and LCD/LED displays keep it lit bright as day 24/7.
Down at the southern tip (skipping a bit), you've got a variety of smaller neighborhoods. SoHo, the Village, Little Italy (which is half taken over by the expansion of Chinatown), etc. Each with its own character.
Skip South Street Seaport, BTW. There isn't much to it, and there are better things to do with your time than walk around a glorified mall.
Down past that, you've got the Financial District, which includes Wall Street and the WTC site.
And then you've got the Statue of Liberty and Ellis Island, accessible by ferry, which leaves from the southern tip of Manhattan. If you book your tickets online at least 24 hours in advance, you can get a pass which will allow you into the museum in the pedestal. Unfortunately, due to security concerns, you can no longer climb the statue itself, but you can go to the top of the pedestal to the bottom of the staircase where you can look up her skirt. That is, see, through a couple of plexiglass windows, part of the inside of the statue.
As far as areas to avoid... Harlem starts at about 120th Street. The area is being "gentrified," but still might not be the best place for tourists. Also, the night life in some of the areas south of Houston can get a little wild. Not necessarily unsafe, but not as... touristy as the daytime.
There are also a wide variety of tours available. Walking tours, bus tours, water tours, a "duck" tour (not official Duck Tour, but in a "duck" - converted WWII amphibious vehicle which acts as both bus and boat).
And that should keep you busy enough that you don't need to worry about the Outer Boros just now.
Oh, one more "landmark" which you may not find in the tourist guides:
238 East 72nd Street (near 2nd Ave) is a modest little apartment building which happens to be the filming location of Carrie Bradshaw's front stoop. If you're into that sort of thing.
We're going to visit a language school. But for our free time we'd like to hear what we should see and what we should avoid. Any advice from anyone who lives there or has been there is welcome. 
