Queens is a crescent-shaped borough traversing the width of Long Island and including two of the major New York City area airports, LaGuardia (LGA) and John F. Kennedy International (JFK). 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, investors held the 1939 and 1964 World's Fairs. 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 Flushing Meadows-Corona Park. The northern end includes Shea Stadium and the USTA Billie Jean King National Tennis Center; 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.
For information on how to walk or bicycle to and from Queens, check http://www.transalt.org/ . Except for the Whitestone and Throgs 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 Airport. A proper tour of Queens is worthwhile. It can be conducted by a stalwart driver, as the roads can be tough to navigate. Much of Queens (but unlike Manhattan, not all of it), including many of the most interesting parts, can be seen by subway. A trip on the 7 train, made nationally famous by the contempt of former Atlanta Braves pitcher John Rocker, is a cultural experience in and of itself. The 7 runs elevated through most of Queens, so you'll be able to get a good sense of much of the borough through its windows. A good tour of Queens should include at least three meals in three different ethnic enclaves.
Other subways for getting around (and in and out of) Queens include the A, E, F, G, M, N, R, V & W. The Long Island Rail Road makes several stops in Queens: the main line runs through central Queens and the Port Washington line runs along the north shore (including a stop in Flushing).
Queens is quite diverse in density and character. While western Queens (closer to Manhattan) is urban, much of eastern Queens is relatively suburban. As 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. Rising 50 stories, the building, the result of Citibank's attempt to create a new business district in LIC, is the tallest building in New York State located outside of Manhattan.
A number of museums are located in Long Island City, including the Isamu Noguchi Sculpture Museum in Noguchi's former sculpture studio, the Museum of African Art, Sculpture Center , and the Museum of the Moving Image which includes interactive exhibits on the history of video games. The area also includes two free places to view art, Socrates Sculpture Park which overlooks the East River (next to Price Costco on Vernon Blvd.), and the Fisher Landau Center showing a private collection of contemporary art.
(A general tip on NYC Museums: 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.)
Across the street (Jackson Avenue) from PS.1 is a fascinating site as well: 5 Points which is one of the few "legal graffiti zones" in New York City. Visit the website for an advance taste. The entire building is decorated (including the inside if you can, ahem, find a way in). 5 Ptz is underground New York at its finest--although artists must apply for permits from a shadowy figure with e-mail addresses posted on-site (which perhaps ensures that high standards for the spraypainted art). Few taggers have defaced the site with their idiotic scrawls; rather, the art is better, fresher, and more colorful than many PS.1 exhibits. Be sure to walk around the entire length of the building. Just under the 7 train, which runs overhead on 5 Ptz's north face, you'll find a large enclosure for truck loading, which features some of the best artwork of all. A fire escape runs up to the roof, and of course, every space within arms' reach is decorated as well. If you're into this stuff, you'll want to bring a camera.
In Flushing Meadows Corona Park (also on the 7 line; exit at Shea Stadium), the Queens Museum offers visual art, cultural events, Worlds' Fair Memorabilia, and a sprawling scaled-down Panorama of the entire city. It's incredibly accurate --except they've yet to remove the World Trade Center.
Just off Northern Blvd. in the area between Astoria and L.I.C. --at 35th Avenue and 36th street --you'll find the Museum of the Moving Image, which showcases movies and the televisual arts, including video games, with revolving exhibitions. Kaufman-Astoria Studios (home of the Sesame Street, among others) stands next door; there's also a gigantic movie theater, and a nice new 24 hour diner/bar (which serves pitchers of beer) known as Cup. Take the R/V/G or the N/W line.
Note: Jacob Riis is a federal beach and subject to different laws than the rest of the Rockaways.
Visit farm markets at:
Similarly to all of the other borough of New York City, Queens has many distinct neighborhoods. See the eat section below for more information on Queens neighborhoods.
As New York City is the birth place of hip-hop culture, there are hundreds of records stores scattered around the boroughs, and some are in Queens. Also, though vinyl has disappeared from the shelfs of regular record stores, many stores still sell used and new vinyl.
Snobbish Manhattanites never come to Queens, which is one of its great appeals for those who live there. There are a few top-notch bars in Queens, but it's the restaurants that really shine, for a simple reason: Manhattan food is Yuppie food; Queens food is aimed at genuine ethnic inhabitants. To put it another way, come here if you like spicy food. If you want a real taste of Hong Kong--or Tibet, Indonesia, Colombia, Peru, India, Argentina, or just about anyplace (including France)--you'll find it in Queens.
Suggested general areas for culinary roving:
If you like bubble tea with sago and tapioca, there are several good spots within a few blocks of the Flushing - Main St. stop on the 7 train. One of the best is on the corner of Main St. and 39th Av.
Queens is home to one of the most entertaining and pleasant places to sip a brew, the Bohemian Hall (known citywide simply as "The Beer Garden"). Drink Czech ales by the pitcher at wooden picnic tables under leafy canopy, surrounded by hundreds. 29-19 24th Avenue, just west of 31st Street. N/W to Astoria Blvd. 
There are a number of hotels in Flushing that serve LaGuardia Airport, including a Sheraton. There are also many hotels near Kennedy Airport in Jamaica, but the location is generally considered undesirable for visitors, except for its proximity to the airport. Some hotels in Jamaica are listed as three stars but are nevertheless poorly kept. Other hotels are scattered through Kew Gardens, Forest Hills, Elmhurst, Long Island City, and various other neighborhoods.