Queens is a crescent-shaped borough traversing the width of Long Island and including both major New York City airports. It also carries the largest ethnic diversity in its area of any region in the world, divided into small enclaves. Jackson Heights, for example, includes a huge Indian area, followed by a Colombian area, and then a Mexican area. Each offers a wide array of authentic shops, native-style cuisine, and festivals modified only slightly by the generally colder New York City experience.
The geographical center of New York City is actually in Queens. Near this location, the 1939 and 1964 World's Fairs were held. The area still includes an interesting museum and some architectural and artistic relics of the events (including the Unisphere, a 300 ton spherical grid of steel sculpted to look like the globe--as seen in Men In Black). The area is now called Corona Park. The northern end includes Shea Stadium and the U.S. Tennis Open stadium; further north still one can walk along the edge of a marina in Long Island Sound. The park also includes a science museum, a zoo, pedal-boats, and frequent special events.
One can also visit the beach without leaving Queens--as the Ramones put it, "we can hitch a ride to Rockaway Beach." Rockaway Beach is actually a fine stretch of relatively unpolluted sand on its own island linked by two bridges to Long Island. Aside from the still-visable Empire State Building, and the frequent planes landing at nearby John F. Kennedy Airport, it is perhaps the most pastoral experience in New York City. The cleanest beach can be found at Gateway National Park (also known as Jacob Riis Park). Note that women CAN go topless at any public New York beach, though not very many do.
Much of Queens is relatively suburban--in every borough, the closer you get to Manhattan, the more rare it is to find a stand-alone house. The more urban clusters are in the northwest: Astoria and Long Island City (LIC). LIC also contains Queens' only skyscraper, the "other" Citibank building, located directly across the East River from the more prominent angled-roof skyscraper in Manhattan.
Queens is currently the home of the Museum of Modern Art, which houses well-known masterpieces by Van Gogh, Dali, Rousseau, Oldenburg, and their 20th-century ilk. Eventually the MoMA will return to Manhattan, but in the mean time, stop by. The tickets are not cheap, but the art is worth it. (If you work for a large company such as IBM, GE, or Citigroup, check to see if your company is a member--this goes for all museums in NYC; different museums have different sponsors of course.) Near the museum one can also find the Isamu Noguchi Sculpture Museum and the Museum of African Art (in the same building).
For information on how to walk or bicycle to and from Queens, check http://www.transalt.org/ . Except for the Whitestone and Throg's Neck bridges, all the bridges can be crossed by pedestrians and bicycles. Be prepared, however, for long walks--and don't forget that Queens is very, very big--and not well-designed for a walking tour. Do not attempt this without a map!
Sadly, most Queens visitors spend their visit on a bus to or from LaGuardia Airport or JFK. A proper tour of Queens is worthwhile. It should be conducted by a very stalwart driver (the roads are hellish to navigate) and include at least three meals in three different ethnic enclaves. One should also not miss the MoMA.