“I’ll take Manhattan, the Bronx and Staten Island too!” – the words of the evocative old tune may have left out two of the Big Apple’s other boroughs – Brooklyn and Queens – but at least it acknowledges that there is more to America’s most cosmopolitan and vibrant city than Manhattan.

All too often visitors limit themselves not just to Manhattan but to its Midtown, home of the iconic Empire State Building, Rockefeller Center, Radio City Music Hall, the glittering theatre land around Times Square and posh, shop-lined Park Avenue.

I always try to head farther afield as well – perhaps to the tranquil Cloisters: Medieval Art and Architecture museum at the northern tip of Manhattan Island, or to the southern tip for the ferry to the Statue of Liberty, Ellis Island or Staten Island.

The once-gritty Meat Packing District, where I used to stay with friends in a Jane Street town house, has been transformed. Gone are the meat carcasses hanging from butchers’ shop fronts and the colourful diners with their equally-colourful clientele. In their place – the extension of the award-winning High Line park and walkway, built along the bed of a former elevated train track, and the new, cutting edge Whitney Museum of American Art.

Then there is the rebirth of the World Trade Center, destroyed during the terrorist attacks of September 11, 2001. Its One World Observatory crowns what is now the tallest building in the Western Hemisphere; nearby are the thought-provoking 9/11 Memorial & Museum and new Westfield shopping and culinary complex.

Take the ferry to Ellis Island to view the fascinating new exhibitions in the recently-renamed and revamped Ellis Island National Museum of Immigration or to Staten Island for Snug Harbor’s new Staten Island Museum. Next year, the island’s attractions will include Empire Outlets, New York City’s first and only discount outlet shopping centre, and, by 2017, it will be home to the world’s largest (650ft tall) observation wheel, considerably outflanking the London Eye (443 feet).

The New York Aquarium will open next year in Brooklyn’s colourful, revitalised Coney Island, where you can continue to enjoy the amusement park rides and wide, sandy beach. I also love visiting that buzzy borough’s beautiful Prospect Park, treasure-filled Brooklyn Museum, elegant Brooklyn Heights, with its stunning views of Manhattan, and the quaint cobble-stoned area of restaurants and shops tucked away under Brooklyn Bridge.

Visiting friends in Queens’ Long Island City, I discovered the newly-expanded and renovated Sculpture Center and Socrates Sculpture Park. Meanwhile, in Astoria, the Museum of the Moving Image has opened its new Jim Henson Gallery, featuring the puppets, props and costumes from such shows as Sesame Street and The Muppets. And jazz fans can visit the home of Louis Armstrong, one of a number of famous musicians who resided in Queens.