As shafts of fading autumn sun dance through a bank of clouds above the towering Washington Monument, my eyes are drawn eastward to the Statue of Freedom sitting high atop the Capitol Building.

David, my affable guide from Wisconsin, points to the 19ft bronze female figure that embodies the founding spirit of the United States.

“She faces east because the sun sets in the west,” he explains.

“So as long as she stands there the sun will never set on the face of freedom.”

My bicycle tour of Washington’s monuments and memorials is coming to an end and the observation merely reaffirms what became so apparent as I pedalled from one stop to the next through the leaf-strewn paths of the National Mall.

“There are no coincidences in DC,” David adds, flicking up his bike stand as he prepares to head back to base at Unlimited Biking.

“There’s a plan behind everything.”

Symbolism pervades the US’s capital city, and no visit is complete without a tour – by foot, bike or bus – of the familiar landmarks that chart the story of a nation through narrative architecture.

Most of them are centralised and relatively close together, so you can easily take in all the main monuments in one afternoon, with enough time to fit in the White House and Capitol Building along the way.

My highlight is standing on the granite steps of the Lincoln Memorial, looking out on the shimmering waters of the Mall’s vast reflective pool from the very spot where, in August 1963, Martin Luther King Jr delivered his era-defining ‘I Have a Dream’ speech.

Given its storied past, the challenge for Washington has always been widening its appeal beyond its rich heritage and attracting visitors keen to see something new.

The freshly reopened National Air and Space Museum is a fine example of how the city is attempting to balance its past with its present.

The museum is midway through a $1 billion refurbishment and the first completed section has just recently been unveiled to the public.

The collection houses some remarkable items, from the Wright brothers’ Flyer, which completed the world’s first manned flight, to the Columbia command module that brought the original lunar astronauts back to earth. Neil Armstrong’s Apollo space suit is also on display.

For a venue that has welcomed 350 million visitors since it opened in 1976, the task facing designers was to showcase the history of flight for a new generation.

“These are stories we’ve always told, but now we’re telling them in a way that’s meeting the 21st century,” curator Jeremy Kinney tells me, as he proudly lists the high-tech interactive features that begin to explain the colossal refit bill.

More than half of the 1,200 artefacts in the new section have never been exhibited before, including a full-sized X Wing fighter from the Star Wars franchise.

The museum is free, and for that you can thank an English chemist who never set foot in the United States.

James Smithson, who died in 1829, bequeathed $500,000 (£438,722) in his will to establish an institute in Washington that would promote the “increase and diffusion of knowledge”.

He had one stipulation – that entry would be forever free.

16 of the 19 Smithsonian museums are located in Washington. It would take years, never mind days, to take in all the exhibits, so you need to be selective.

As well as the Air and Space Museum, I visit the National Archives – where the Declaration of Independence, Constitution and Bill of Rights sit side-by-side – and the National Museum of Natural History, where giant model dinosaurs, whales, elephants and sharks provide a ‘wow’ factor for young and old.

Another recent addition to the Smithsonian collection is the National Museum of African American History and Culture. It is the only such museum in the US dedicated exclusively to the African American story and is well worth seeing.

I have the fortune of being able to combine another Smithsonian favourite – the American Art Museum – with a dining experience like no other.

One night a year, the museum hosts the Embassy Chef Challenge – a charitable initiative that offers a unique opportunity to sample food from around the world, courtesy of the city’s 175 diplomatic missions.

As I stand beneath the opulent glass canopy of the museum’s central courtyard, sipping a Norwegian cocktail while sampling a Kazakhstani rice dish, one thought recurs: ‘Only in DC…’.

Outside of the Smithsonian Institution, the International Spy Museum gives visitors the chance to assume a false identity and take on a secret mission as they journey into the shadowy world of espionage.

Then there is the National Geographic Museum’s ‘Beyond King Tut: Immersive Experience’, which marks the 100-year anniversary of the discovery of Tutankhamun’s tomb with a redolent cinematic assault on the senses.

During my three-day trip to the city, I stay in the Thompson Washington DC, a highly-rated property close to the National Mall, whose well-appointed rooftop bar offers guests bespoke cocktails and sunset views of the Anacostia River.

It is set amid a vibrant waterfront district that has undergone a dramatic transformation in the last decade, with industrial warehouses giving way to a modern mix of shops, cafes, restaurants and workspaces.

My visit coincides with the opening of the latest phase of the major Wharf development on the banks of the Washington Channel.

I could have picked no better spot to watch the celebratory fireworks than my window-side table at the fabulous Japanese restaurant NaRa-Ya.

The popular waterside eatery is one of several restaurants that collectively are propelling DC’s advancing reputation for culinary excellence.

The Grill, which is nearby, offers an American menu loaded with wood-fired seafood and red meat, while the Michelin-rated Bammy’s in Navy Yard is worth a try for the Caribbean-style jerk chicken alone.

The rooftop beer garden and restaurant at Takoda is also popular with diners and revellers alike.

For those needing sustenance on the northern side of the National Mall, the Peruvian gastro bar Pisco Y Nazca Ceviche is highly recommended.

It also provides a good starting point for an evening in the uber cool speakeasy style bars of 14th Street.

Inspired by the Prohibition era, half the fun of checking these out is the challenge of finding them.

The wood-panelled bar at Chicken and Whiskey is accessed through a refrigerator door at the back of a rotisserie diner, while I walked past The Gibson twice before realising an unmarked black door offered a portal to the candlelit cocktail lounge within.

On my final day in Washington, I head to the quaint cobbled street neighbourhood of Georgetown, with a bracing walk by the Potomac River and brunch at the excellent Farmers Fishers Bakers more than enough to blow away the cobwebs from an evening on 14th Street.

It is home to Georgetown University and a morning stroll around the tree-lined walkways of the famous academic campus is a restorative treat.

A bit more daunting are the steepling steps at the nearby corner of 36th Street and Prospect Street, where one of the most famous scenes from the 1973 horror movie the Exorcist was shot.

Unlike the priest who was thrown to his demise down the narrow staircase in the film, I navigate them without incident, albeit needing several minutes to catch my breath.

It was an enjoyable way to end my break in DC.

As I head to the airport for the flight home, I reflect on a quote displayed at the Air and Space Museum from pioneering female aviator Amelia Earhart: “There is more to life than being a passenger.”

It serves as an instructive reminder for all those visiting Washington that there is so much to see and do if you are prepared to go and find it … even if it is behind an unmarked door on 14th Street.

How to plan your trip

For more information about the destination, visit washington.org.

Aer Lingus operates two flights daily between Dublin and Washington DC. Fares start from $229 (£200) each way, including taxes and charges. Visit aerlingus.com.

Room rates at the Thompson Washington DC start at $200 (£175) a night. For more information visit hyatt.com.