You may also like these photo galleries
Inwood Hill Park
Covering nearly 80 hectares at the tip of Manhattan, and flanked by the Harlem River, Inwood Hill Park is a last remnant of what Manhattan used to be like. That means caves, glacier-carved ridges and deliberately unmanicured forest. There are also salt marshes occupied by water fowl, such as Canada geese, while raccoons and skunks can be found running around elsewhere. It's the complete opposite of the Manhattan preconception. See nycgovparks.org.
The Cooper Hewitt Smithsonian Design Museum
Overshadowed by the nearby Metropolitan Museum of Art and the Guggenheim, the Cooper Hewitt celebrates all things design inside a glorious old mansion. There are some hugely enjoyable exhibits – such as an installation on how Pixar films are made or the room full of mirrors with dozens of silver shoes on the floor. But it's also brilliantly interactive, with electronic "pens" that allow you to scan artefacts, then bring them up on giant screens and design hats or chairs based on them. See cooperhewitt.org.
The Rockefeller Centre
The Rockefeller Centre is hardly undiscovered – huge numbers of people head up to the Top of the Rock observation deck. But far, far fewer take the time to look around the complex and plaza, which are full of public art. The main lobby of 30 Rock contains a striking, detailed mural called American Progress, the fountains in the plaza are full of various spitting animals, and the mighty twin statues of Atlas and Prometheus represent strength and civilisation. See rockefellercenter.com.
36 Hours in Central Park
With its endless trails, hidden nooks, museums and nearby night spots, Central Park is that rare tourist destination that is also a pleasure ground for locals. Video by the New York Times.
Easily the quirkiest classical music venue in NYC, Bargemusic is an old coffee barge moored near the Brooklyn Bridge on the Brooklyn side of the East River. The floating concert hall has an intimate, wood-panelled interior, and hops all over the shop in terms of recitals. Performances at 4pm on Saturday are free, but reservations are advised. See bargemusic.org.
The Lower Eastside Tenement Museum
One of the Lower Eastside's old tenement buildings has been saved, preserved and turned into a time capsule. The tours inside it are much less about the building itself than the people who lived there and the way of life for the hard working poor in old New York. The tours become more about storytelling, piecing together the lives of immigrant inhabitants from historical records and family accounts – and there are some surprising twists and turns along the way. See tenement.org.
The Museum of American Finance
Usually ignored because it sounds so dull, the Museum of American Finance tells the story of money in the States, from the first banks set up by Thomas Hamilton to keep the fledgling independent nation afloat to the modern day Federal Reserve system. There are displays of old coins and notes, of course, but the museum is really telling the story of how the US became the world's sole superpower. And that doesn't sound nearly as boring, does it? See moaf.org.
Brooklyn Superhero Supply
The odd store name turns a few heads, but head inside and it's one of the weirdest shops you're ever likely to counter. You can buy omnipotence or telekinesis in jars, stock up on cybernetic henchfish or stand in a cape-testing machine. This shop for would-be superheroes is actually a front for a children's after-school tuition programme, but a wonderful degree of creativity has been applied to the whole concept, and it's easy to get happily lost in the detail. See superherosupplies.com.
The Hudson River
There are plenty of cruises and boat trips that go along the Hudson, but a hugely underrated way of experiencing the city is in a kayak, dodging between ferries and getting a clear view of the skyline from low down. The Manhattan Kayak Company (manhattankayak.com) offers both rentals and private tours from Pier 66, with the combo of sun going down and city lights going on making the sunset tour special. See manhattankayak.com.
A long, narrow island in the middle of the East River, Roosevelt Island is skipped over by the Queensboro Bridge. It's largely residential, but at the southern tip there's the rather lovely Four Freedoms Park. Created as a memorial to Franklin D. Roosevelt. It is designed to be like a simple, roofless Greek temple, and offers spectacularly framed views of the Manhattan skyline. But 90 per cent of the fun is getting there by cable car. See rioc.ny.gov.
The New York Public Library
A lot of people walk past the library system's main building in Bryant Park, but few venture in. That's a mistake, because inside the hulking marble colossus is a treasure trove of fascinating old maps and important old books. The free guided tours are the best way of uncovering the library's hidden-in-plain-sight secrets. See nypl.org.
See also: The 20 things you must do in New York