“Of all the gin joints in all the towns in all the world, she walks into mine,” said Humphrey Bogart in Casablanca.

Except you'd be out of luck if you were looking for a gin joint in Fjällbacka, the Swedish former fishing village where Ingrid Bergman spent her summers after buying the nearby island Dannholmen in 1958. The nearest state-run alcohol shop (anything over 3.5 per cent proof is monopolised by the government) is seven miles away, although the town's only hotel can fix you something stronger.

Bergman’s ashes are sprinkled in the sea in this gorgeous archipelago, and her presence lives on in Fjällbacka: her sculpted head is mounted on a plinth on the promenade, alongside three boards displaying large black-and-white photographs of her.

Here is Bergman trying to blend in while buying vegetables at Fjällbacka's street market; here she is looking a bit peeved as a photographer snaps away as she strides through the village. The more relaxed pictures of her at home on her private island – bought with third husband, Swedish theatre producer Lars Schmidt – show her soaking up the sunshine alongside the children of her second marriage, to Italian film director Roberto Rosselini: Robertino and Isabella.

Dannholmen is still in private hands, but you can catch a ride on the water taxi to several other islands off the coast of Fjällbacka. For 40 kroner (about £4) I made the short hop to Valon, where a well-marked walking trail though woods and past granite boulders will lead you up to a vantage point taking in the whole Bohusian coast – the stretch between Gothenburg and the Norwegian border.

Käringön, part of the archipelago of pretty islands floating off the coast of Sweden (Inntravel)

Bohuslan was part of Norway until the 17th century, and apparently a Norwegian twang can be detected in the local accent. Inntravel – which usually specialises in walking holidays – has chosen this corner of western Sweden as one of the destinations for their inaugural driving tours, which launched last year, and it doesn't take long to see why they’ve made the exception.

The landscape is by turns intimately bucolic – red wooden farmhouses (the rust-coloured paint is known as "Falun red" after the Swedish copper-mining town where it was first extracted), pine and birch trees, granite outcrops – and dramatic, with sudden vistas of oceans, lochs and fjords appearing as you drop around another sharp corner. It’s not as dramatic, admittedly, as in Norway, but it’s still pretty stirring. You take the long, twisty route southwards, by way of the many islands that are linked by either bridge or ferry. The ferries are free, and you might even be the only passenger.

Being walking specialists, Inntravel also suggest hiking itineraries for when you want to get a little deeper into the landscape – whether that’s scrambling through the heather and stunted holm oaks on Valon island, or navigating Gothenberg’s busy streets. After all that rural beauty, Sweden’s second city – about 100 miles from our starting point at Fjällbacka – is a welcome urban break.

The roadtrip starts in Fjällbacka, 100km north of Gothenburg (Inntravel)

Inntravel's self-drive itineraries – so detailed that it even suggests pitstops for lunch – are topped and tailed by Fjällbacka and Fiskebackskil, another former fishing village now colonised by second-home owners, who’ve tarted up the traditional wooden houses in a spectrum of colours instead of the standard Falun red. It’s a quiet place – there are hardly any cars along the narrow cobbled lanes, and hardly any people, either. A large dog eyed me warily as I walked past.

More ambitious island-hopping suggestions are to the Weather Islands – 30km from the mainland and the most westerly islands in Sweden – and the protected Koster Islands, where you can rent bikes. Elsewhere, there are “seafood safaris” (boat trips to a mussel farm) and kayak hire (there are no strong currents along the Bohusian coast, making it a relatively safe area for boating and swimming). I stopped off at the Vitlyke Bronze Age Museum near Tanum – the Bohusian region is well known for its Bronze Age carvings, and many are on display here, along with a replica farm.

But the real highlight of this tour is the open road, winding its way along the coast on almost empty and well-maintained highways. It's quite a shock to the system when you eventually join the E6 motorway to speed you back to Gothenburg, and the 21st century.

Travel essentials

Getting there

Inntravel’s week-long self-drive Sweden’s West Coast trip starts from £1,170pp, including seven nights’ accommodation on a B&B basis, five dinners and five days’ car hire. It’s available 1 June–30 September. BMI Regional flies direct to Gothenburg from Birmingham, Norwegian flies from Gatwick and British Airways from Heathrow.