Eagle-eyed football fans heading to Berlin’s Olympiastadion for the Euros may spot what looks like an abandoned space station topped with four enormous white orbs on a hill a mile or so south of the ground.

The stadium – which will host the Euro 2024 final – has a more storied past than perhaps any other. Built by the Nazis for the 1936 Berlin Olympics, the great African American sprinter Jesse Owens won four gold medals here in front of a watching Adolf Hitler, single-handedly debunking the Führer’s myth of Aryan supremacy.

But the weird sci-fi-looking building nearby has an even more fascinating history. Teufelsberg (Devil’s Mountain) is a manmade hill eight miles west of the city centre, created from 25 million cubic metres of second world war Berlin rubble that was piled on top of a Nazi military academy to form the highest point in the city. Then, at the height of the cold war, US intelligence agencies built a hilltop listening station on Teufelsberg to intercept East German and Soviet communication. The station’s towers and enormous antenna radomes – those white domes that resemble giant golf balls which protected the radars – are almost comically eerie.

I’m in Berlin with my 15-year-old son, George, and we take the S-Bahn out to Grunewald, then hike up through the forest to Teufelsberg. After German reunification, the hill fell into disrepair and is now a weird and wonderful canvas for street artists.

We climb the listening station tower to take in stunning views of the Berlin skyline and what is one of the world’s largest street art galleries, with more than 400 murals by artists from all over the world. It feels like an abandoned set from Dr Strangelove that’s been reclaimed by graffiti artists. George thinks the whole place is “very cool but a bit bonkers”.

We’re in Berlin to explore the city’s enthralling cold war legacy. Like most teenagers, George has to be dragged around ancient landmarks, however momentous (on a recent trip to Pompeii he declared himself officially bored 30 minutes in), but loves exploring anywhere with a more relatable modern history. (Two of our best trips have been on the Beatles’ trail in Liverpool and a tour of political murals and the Bloody Sunday march in Northern Ireland.)

So where better to take a teenager than Berlin, epicentre of 20th-century European history? As well as kid-friendly favourites like the Wall and Checkpoint Charlie, there is lots of “new” cold war history to explore. And equally important for a teenager, it is one of the coolest cities in the world, packed with great shops, markets and “edgy” neighbourhoods.

We’ve come on the train from London, and as the overnight sleeper pulls into Gesundbrunnen station, it is still dark, deserted and shrouded in a pre-dawn mist – such a broodingly atmospheric welcome to the city, it feels made to order for our cold war adventure.

The European Sleeper we’re on (it starts in Brussels and was recently extended to Dresden and Prague) feels like an Eastern bloc relic, too. The carriages were built in the 1950s and 60s and are full of Soviet era “quirks”: the doors of the compartment don’t fully close, the heating is on full-blast all night, and the curtains look as if they went up before the Wall. But the duvets are soft, the berths comfortable and we’re rocked to sleep as the train trundles through the night from Amsterdam to Berlin. (For travellers who would rather connect in Paris, and travel in something less retro, a new Nightjet sleeper route links the French and German capitals.)

On our first day we take in the Reichstag and walk through the mighty Brandenburg Gate on the way to the new Cold War Museum, which brings the era to life through a series of gripping virtual reality tours. The first takes us back to the seminal moment when Berlin was literally split in two, right down the middle of Bernauer Strasse, which we’re standing in here. The story is told through the eyes and words of a terrified East German soldier; we hear him agonising over his future in a divided Germany: should he stay or go? In a heart-stopping moment, he deserts his post and leaps from east to west over the barbed wire border and into another country.

Nowhere in the city emanates history like Tempelhof airport. Designed by the Nazis to be the world’s largest airport and a showcase of power, it was never completed, but still played a pivotal role in both the second world war and the Berlin airlift in the late 1940s. If we’d stood here in 1948, the air would have been thick with American, French and British aircraft (a plane landed here every 45 seconds at times) airlifting food and fuel into West Berlin after Stalin had blockaded the western outpost.

To celebrate the centenary of its 1923 opening, an elegant wooden terrace has been added, giving access to the airport’s original control tower and views of the city centre. Since the airport closed in 2008, the runways have been turned into Tempelhofer Feld, one of the world’s largest city parks, and we look out from the control tower on hundreds of cyclists, skaters, runners, dog walkers, revellers and families enjoying picnics and barbecues.

Like Teufelsberg’s ever-evolving canvas, the unfinished airport still feels like a work in progress. This is what makes Berlin so fascinating: what to do with all this living history as the city figures out where its difficult past sits in relation to its uncertain future? It’s why four-metre-tall sections of Berlin Wall are still strewn about the centre: some curated in outdoor galleries or left as they were in 1989; other standalone sections are now memorials or tourist attractions; we even spot one in a junkyard, awaiting a new home.

Even our hotel has a cold war legacy: you wouldn’t guess it now, but the lovely Lux Eleven apart-hotel (double rooms and apartments from €106), on vibrant Rosa-Luxemburg-Strasse in Mitte, was once home to the Soviet secret service. From our window, we have a neck-wrenching view of the famous East German TV tower, a reminder that we are in the former GDR.

But east Berlin isn’t all about the cold war. On Karl-Marx-Allee, the showcase boulevard built by the communists after the war, the Computerspielemuseum documents the history of arcade computer games. George joins the dozens of other excited kids discovering the communal joy of 20th-century video games, including classics such as Asteroids, Space Invaders and the original Playstation.

There’s even an arcade machine developed in the GDR in the mid-80s: Poly-Play contains seven different games, including a Pac-Man clone. Some of the 300 games that can be played are housed in an original 1980s video arcade. Like the Berlin Wall, a 20th-century phenomenon that has been consigned to history.

• The trip was supported by Visit Berlin. Rail transport from London to Amsterdam was provided by Eurostar (from £39 one-way) and from Amsterdam to Berlin by the European Sleeper (couchettes from €49 one-way).