You can't go wrong with a river, especially when planning a walk across a sprawling megacity like Seoul, home to about 24 million people – half of South Korea's entire population. The mighty Han river flows east to west through the centre of the South Korean capital, splitting the city north and south, as the Thames does in London. In a city where many of the streets have no name, and buildings are numbered according to the order in which they were built (rather than appear), following the river makes good sense – especially as a 40km cycling trail runs alongside it.

So my partner Nick and I map out a five-day walk from Seoul's eastern fringes: we will wend our way westwards along the river, via a place in Itaewon we hear does killer martinis, and with a small detour north to Mount Namsan. Look at any map of Seoul and you'll see the 262m peak in the middle of the northern sprawl, its juxtaposition to the city rooted in ancient feng shui lore – the city was constructed around the mountain to create a geographic harmony.

It's shockingly cold as we set off down the hill from our hotel, in the aptly named suburb of Walkerhill, in the north-eastern district of Gwangjin. Walkerhill is a tasteful sprawl of hotels, universities and hospitals on the edge of Seoul's metropolitan area, and a logical place to start. We should have had an amazing view of the Han and the city beyond, but everything is shrouded in bright white fog. We can hear but not see the busy Gangbyeon Expressway below, along with the Olympic Highway, the main road in and out of Seoul, on the south bank of the Han.

The hotel bus pulls over and asks if we need a lift into Seoul? "No," we reply, with purpose. "We are walking."

The door wooshes shut and the bus roars off, windows framing the faces of the staring Japanese shoppers inside. Staring, because perhaps they have, like us, just arrived in Seoul and their brains are soaking up everything they can see, eager to cut the intimidating reality of a new city into bite-sized chunks of opinion, experience and landmarks. Or perhaps they just think we're crazy.

We walk briskly to stave off the cold, and to dispel rogue thoughts about how much warmer and easier it would have been to hop on that bus. It's early and there aren't many people about. We pass an empty petrol station, a motorbike garage, a couple of noodle shops with steamy windows, a small grocery, more noodle shops … When we come to a supermarket I race inside, scanning the shelves of tins and packets. Clothes shopping leaves me cold, but I can wander for hours around foreign supermarkets: strangely-named crisps, bottles of ginseng, scary-looking cuts of meat. Supermarkets are the museum gift shop of a nation's culture.

Back outside, we wait to cross the road. A small, neat Korean man in a black puffa jacket waits too. Mr Park is from Daegu, a city 200km south of Seoul, and is walking from the same hotel. "I come here often for business but like to start each day walking with my thoughts," he says, smiling shyly: "Every day, exercise and contemplation: it makes a life happy."

Mr Park speaks excellent English, having lived in the US while in his twenties. "Much of our society is based on the American model: we shared its culture and learned from its influence after the war."

I hadn't expected the Korean war (1950-53) to come up only an hour into our first day. Seoul has been a centre of culture and influence since the 1300s. But the city also has a history of invasion and occupation.

At Gwangnaru, one of the stations on Seoul's cheap, extremely efficient underground network, Mr Park courteously wishes us a good day and disappears down into the underground. Nick and I walk on to the commuter hub of Gangbyeon, where we plan to head south across the river on the Jamsil bridge. It's busy now, and we're caught in the stream of commuters, swathed in the black padded coats which we're starting to recognise as Seoul's winter uniform. Crossing a complicated intersection, we're swept past a Buddhist monk who's kneeling in prayer on one of the small traffic islands.

The pavement outside the busy rail and underground station is lined with ramshackle stalls selling magnifying glasses, socks and skewers of barbecued chicken. We walk up a ramp onto the wide Jamsil bridge, and finally, here it is, the Han, a flinty grey carpet a kilometre wide. Our eyes travel along then beyond it, to the city in whose shadow it flows. Seoul's a monster: an unflinching skyline of serrated high-rises, stretching into the distance like battlements.

Twenty minutes later we're still walking across the bridge, and it feels like we haven't progressed an inch. The cyclists on the adjacent bike path have no such shortage of energy or forward motion. Swathed entirely in black, their faces hidden behind plastic face masks and helmets, they fly at us like angry insects.

On the south side of the river, a tangle of busy roads leads to quiet streets of grocery stores and elegant patisseries, the first clue that we're on the edge of Gangnam, one of Seoul's richest neighbourhoods. Over the next hour, mothers walking children to school are replaced by black puffa-ed office workers. Little shops grow bigger, streets grow wider and office blocks rise ever higher. And finally we arrive at our planned coffee stop, the Coex, which is the biggest shopping mall in Asia at the World Trade Center on Samsung-dong.

Coex is an embodiment of Seoul's own riches to rags and back to riches history. Rapid industrialisation (dubbed the Miracle on the Han) after the devastation of the Korean war saw Seoul transformed into a fully-fledged economic and technological superpower. Construction projects boomed along with Seoul's fortunes, and by the 1970s, the city was in the grip of such building fever that developers were driven underground in the search for affordable space.

The result is a series of vast underground shopping malls, subterranean emporiums beneath Seoul's traffic-choked streets. And the mammy of them all, Coex, covers 88,000 square metres. There are shops, food courts, cinemas … and even a 90-tank aquarium, complete with sharks, rays and penguins.

We experience it all in a jetlagged stupor, stumbling against the tide of bodies along strip-lit avenues filled with Body Shop, Zara and Nike stores. Cut off from natural light in air-conditioned avenues smelling of popcorn and soy, it's like being in Vegas. You could lose days in here.

Back outside, the cold air slaps us awake, and we thread our way through Gangnam: the shopping district made famous by that song. Shoppers seem younger, thinner and more colourfully dressed on Brand Name Street. The twisting back alleys might as well be called No Name Street for all the signs we can find. But all we're looking for is the river so, going old school, I break out the compass and follow it north.

Twenty-seven bridges span the river Han in downtown Seoul. As we cross in the gathering dusk, going north towards Itaewon, the Banpo bridge upriver is transformed into a spectacular light show. On either side of the traffic-clogged lanes, plumes of water, lit lilac and blue, shoot six metres into the river. Families and young couples watch from the river bank as we squeeze past.

Itaewon's reputation as Seoul's party-hard red-light district stems historically from the US troops stationed at nearby Garrison Yongsan, who used it as an R&R playground. But these days Hooker Hill is a tourist attraction and Itaewon is known as one of Seoul's most cosmopolitan nightlife spots. On this bitterly cold Monday night, however, the long Itaewon-ro main road is deserted. In moodily lit tapas bars and Argentinian fusion restaurants, couples huddle away from the chill seeping through the huge glass windows. We find our much-prized martini bar … but not our martinis. Swaying on high stools at the bar we watch the owner and three bar men anxiously watch a "how to make a martini" video on YouTube. We don't stay till the end of it.

Instead, we check in at the trendy IP Boutique hotel (737-32 Hannam-dong, Yongsan-gu, +82 2 3702-8000, ipboutiquehotel.com, doubles from £130 a night) down the street. Exhausted from the walking, Nick sits on one of the swings hanging from the lobby ceiling and promptly tips over backwards, crashing loudly to the ground. A small dog on the lap of a woman on the next swing releases a volley of tiny, furious barks. It's time for bed.

The next morning the sun is shining, the sky is blue and we walk north into huge Namsan Park. Grassy parklands lead to tree-lined paths and gentle hills; families walk and pensioners do calisthenics on metal apparatus. On the edge of this metropolis there's an incredible sense of nature, and nature being enjoyed. But we only have eyes for Mount Namsan. You can see it from all over the city, but here it is before us in all its misty majesty. On its top, the equally impressive Seoul Tower, which houses an observatory, rears a further 236m into the sky, pointily futuristic.

In our impatience, we cheat and ride the cable car to the top of the mountain, and as we ascend the city slowly unfurls. Above the crush of high-rises, for the first time we see the indentation of streets and the lazy W shape of the river. From the top of the tower, Seoul reveals itself to be built into a series of mountains: a line of glistening skyscrapers wind, like a frozen river, through craggy passes, as far as the eye can see.

Euphoria makes us opt to walk back down the mountain: a decision we regret. For the rest of the walk our calf muscles shriek with each step along the packed shopping streets of Myeong-dong (like Oxford Street, but doing a roaring trade in snail gel moisturiser), through Namdaemun market on Namchang-dong, where I get a pair of cheap specs made. On Sinchon's surprisingly dismal Wedding Dress Street in Ahyeon-dong neighbourhood, we marvel at shop after shop of audacious frocks, which strain at the windows like angry tropical birds.

In the smart alleys behind Ewha Women's University, we drink coffee with serious students, and in the funkier Hongdae University district, EatyourKimChi bloggers Martina and Simon take us to the Vinyl bar on Seokyo-dong in Mapo-gu, where cocktails are served in IV bags and there's scrumptious carrot cake ice-cream on offer.

When we cross back over the river at Yoido, Seoul feels less intimidating than it did last time we were south of the river. We have done it – broken the city into bite-sized chunks. For a peaceful end to the walk, we take a pew in Yoido's huge Pentecostal church on Yeoeuido-dong. It's staggering – like the UN of prayer – as the 10,000-strong congregation sings along with a massive gospel choir. Suddenly, a new, slightly bewildering experience. But thankfully one we're sitting down for.

• Flights were provided by Korean Air (020-7495 8641, koreanair.com), which flies from Heathrow to Seoul from £649 return. Further information on Seoul and Korea: visitkorea.or.kr