There was a buzz in the air on Platform One as crowds gathered, cameras poised, eagerly anticipating the arrival of a rather special grande dame.

Suddenly, in a cloud of steam, the Duchess of Sutherland was making her entrance, working the crowd of excited onlookers as if gliding along a red carpet.

Sporting splendid BR green livery, the Duchess was the steam locomotive for our day trip on the Scarborough Flyer. Built by the LMSR in 1938, she had a busy working life, hauling the country’s fastest trains, before her withdrawal in 1964.

The Scarborough Flyer is operated by the Railway Touring Company, which runs trips on beautifully restored vintage steam-hauled trains around the UK and overseas.

We boarded the Scarborough Flyer at Brighouse – it had left Crewe earlier that morning, picking up passengers at Stockport and Rochdale before travelling through the Calder Valley – and followed the route of the old Lancashire and Yorkshire Railway to Wakefield, then on to York.

Sinking into comfortable seats in a lamp-lit first-class carriage, we were served pastries and coffee as we soaked up the Vale of York views unfolding like a landscape painting. Highlights included lovely Kirkham Abbey and the ghostly-looking old Castle Howard station.

My nephews, Sam, 11, and Jack, nine, had fun peering out of an open window at the steam ahead. With specks of coal dust in their hair, they waved at lone trainspotters standing in fields as we passed through to Malton and Seamer en route to the east coast.

Passengers can visit either York or Scarborough. We ended our three-hour journey at Scarborough’s pretty Victorian railway station with its impressive Baroque clock tower. Passing the Stephen Joseph Theatre, a former Art Deco cinema, now home to Sir Alan Ayckbourn’s productions, we walked into town.

With its North and South bays, separated by a rocky headland dominated by its sprawling Norman castle, Scarborough offers all you’d expect from a seaside resort. Attractions include Peasholm Park; the Sea Life Marine Sanctuary, where ‘residents’ include rescued seal pups; Scarborough Open Air Theatre, Europe’s largest open-air theatre, attracting the likes of Elton John; and the Spa Complex which includes a 600-seat Victorian theatre, home to the only remaining seaside orchestra.

Eighty miles from her Haworth home, Anne Bronte is buried in the town, overlooking the sea. Sick with consumption, Anne was brought to Scarborough by her sister, Charlotte, to take in the fresh sea air, but she died in the town in May, 1849. She was 29, not 28 as it says on her gravestone in St Mary’s Church cemetery.

With the boys nagging for chips, ice-cream and funfair rides, we headed for the South Bay. We ate fish and chips on the beach, bustling with young families sprawled on the sands and dipping toes in the sea, then walked over to one of the lively amusement arcades lining the seafront. Filling plastic tubs with two pence pieces, we tried our luck on the ‘penny push’ machines – I can’t resist the thrill of trying to tip over a pile of coins, winning all of about eight pence – then headed to Luna Park by the harbour.

It’s a delightfully old-fashioned little funfair, with a neon-lit ferris wheel, candyfloss stalls and a helter skelter. We had fun on the Twister and the dodgems, and Jack bounced on the trampolines, then we meandered back along the front.

One of my earliest memories is a holiday in Scarborough, and I recall the little cliff lift tramway running alongside the Grand Hotel.

The UK’s first funicular, the Central Tramway service has shuttled passengers to and from the South Bay since the 1870s and this year it reopened following a £90,000 restoration. The lift first used sea water as counterweight, with two gas engines pumping the water to the upper station to fill the top car, and a bathing pool. In 1910 it was converted to electric and now it’s automatic.

Climbing the cliffside, with splendid views of the bay, I felt a pang of nostalgia for that seaside holiday of long ago, when I sat in the tram car as a wide-eyed four-year-old, clutching my rag doll.

We wandered through town, which has a vibrant mix of gift shops, high street chains and boutiques, and with time running out we returned to the station, where the Duchess was waiting. The North Bay can wait until our next visit.

Tired and sunkissed, we had afternoon tea in our cosy carriage, pulled through one of Yorkshire’s most scenic routes in a haze of steam.